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Continued fractions M¨ obius maps Hyperbolic geometry Toplogical groups Hyperbolic geometry and continued fraction theory I Ian Short 9 February 2010 http://maths.org/ ims25/maths/presentations.php

Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

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Page 1: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry and continued fractiontheory I

Ian Short 9 February 2010

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 2: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 3: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 4: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 5: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 6: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 7: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 8: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 9: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 10: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 11: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 12: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 13: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 14: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 15: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 16: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 17: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 18: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 19: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 20: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 21: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 22: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 23: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 24: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 25: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 26: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 27: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 28: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 29: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 30: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 31: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 32: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 33: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 34: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 35: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 36: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 37: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 38: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 39: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 40: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 41: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 42: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 43: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 44: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 45: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 46: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 47: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 48: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 49: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 50: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 51: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 52: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 53: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 54: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 55: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 56: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 57: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 58: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 59: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 60: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 61: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 62: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 63: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 64: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 65: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 66: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 67: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 68: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 69: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 70: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 71: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 72: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 73: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 74: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 75: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 76: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 77: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 78: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 79: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 80: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 81: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 82: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 83: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 84: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 85: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 86: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 87: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 88: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 89: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 90: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 91: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 92: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 93: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 94: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 95: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 96: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 97: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 98: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 99: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 100: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 101: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 102: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 103: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 104: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 105: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 106: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 107: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 108: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 109: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 110: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 111: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 112: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 113: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 114: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 115: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 116: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 117: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 118: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 119: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 120: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 121: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 122: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 123: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 124: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 125: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 126: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 127: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 128: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 129: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 130: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 131: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 132: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups
Page 133: Hyperbolic geometry and continued fraction theory Iusers.mct.open.ac.uk/is3649/maths/presentations/Wits2010a.pdf · Continued fractions M obius maps Hyperbolic geometry Toplogical

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

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