Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))

Complex Dynamics and

Crazy Mathematics

Dynamics of three very different families of complex functions:

1. Polynomials (z2 + c)

2. Entire maps ( exp(z))

3. Rational maps (zn + /zn)

We’ll investigate chaotic behavior inthe dynamical plane (the Julia sets)

z2 + c

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.



exp(z) z2 + /z2

As well as the structure of theparameter planes.

z2 + c exp(z) z3 + /z3

(the Mandelbrot set)



QuickTime™ and a decompressor


A couple of subthemes:

1. Some “crazy” mathematics

2. Great undergrad research topics

The Fractal Geometryof

the Mandelbrot Set

How to count


the Mandelbrot Set


the Mandelbrot Set

How to add

How to count

Many people know thepretty pictures...

but few know the evenprettier mathematics.

Oh, that's nothing but the 3/4 bulb ....

...hanging off the period 16 M-set.....

...lying in the 1/7 antenna...

...attached to the 1/3 bulb...

...hanging off the 3/7 bulb...

...on the northwest side of the main cardioid.

Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

Start with a function:

x + constant2

Start with a function:

x + constant2

and a seed:

x0

Then iterate:

x = x + constant1 02

Then iterate:



Then iterate:



x = x + constant3 2

2

Then iterate:



x = x + constant3 2

2

x = x + constant4 3

2

Then iterate:



x = x + constant3 2

2

x = x + constant4 3

2

Orbit of x0

etc.

Goal: understand the fate of orbits.

Example: x + 1 Seed 02

x = 00x = 1x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = 11x =2

x = 3

x = 4

x = 5

x = 6


x = 00x = 11x = 22

x = 3

x = 4

x = 5

x = 6


x = 00x = 11x = 22

x = 53

x = 4

x = 5

x = 6


x = 00x = 11x = 22

x = 53

x = 264

x = 5

x = 6


x = 00x = 11x = 22

x = 53

x = 264

x = big5

x = 6


x = 00x = 11x = 22

x = 53

x = 264

x = big5

x = BIGGER6


x = 00x = 11x = 22

x = 53

x = 264

x = big5

x = BIGGER6

“Orbit tends to infinity”


x = 00x = 1x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = 01x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = 01x = 02

x = 3

x = 4

x = 5

x = 6


x = 00x = 01x = 02

x = 03

x = 4

x = 5

x = 6


x = 00x = 01x = 02

x = 03

x = 04

x = 05

x = 06

“A fixed point”

Example: x - 1 Seed 02

x = 00x = 1x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = -11x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = -11x = 02

x = 3

x = 4

x = 5

x = 6


x = 00x = -11x = 02

x = -13

x = 4

x = 5

x = 6


x = 00x = -11x = 02

x = -13

x = 04

x = 5

x = 6


x = 00x = -11x = 02

x = -13

x = 04

x = -15

x = 06

“A two- cycle”

Example: x - 1.1 Seed 02

x = 00x = 1x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = -1.11x = 2

x = 3

x = 4

x = 5

x = 6


x = 00x = -1.11x = 0.112

x = 3

x = 4

x = 5

x = 6


x = 00x = -1.11x = 0.112

x = 3

x = 4

x = 5

x = 6

time for the computer!

Observation:

For some real values of c, the orbit of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

Complex Iteration

Iterate z + c2

complexnumbers

Example: z + i Seed 02

z = 00z = 1z = 2

z = 3

z = 4

z = 5

z = 6


z = 00z = i1z = 2

z = 3

z = 4

z = 5

z = 6


z = 00z = i1z = -1 + i2

z = 3

z = 4

z = 5

z = 6


z = 00z = i1z = -1 + i2

z = -i 3

z = 4

z = 5

z = 6


z = 00z = i1z = -1 + i2

z = -i 3

z = -1 + i 4

z = 5

z = 6


z = 00z = i1z = -1 + i2

z = -i 3

z = -1 + i 4

z = -i 5

z = 6


z = 00z = i1z = -1 + i2

z = -i 3

z = -1 + i 4

z = -i 5

z = -1 + i 6

2-cycle


1-1

i

-i


1-1

i

-i


1-1

i

-i


-i

-1 1

i


1-1

i

-i


-i

-1 1

i


1-1

i

-i


-i

-1 1

i

Example: z + 2i Seed 02

z = 00z = 1z = 2

z = 3

z = 4

z = 5

z = 6

Example: z + 2i Seed 02

z = 00z = 2i1z = -4 + 2i 2

z = 12 - 14i3

z = -52 + 336i 4

z = big 5

z = BIGGER 6

Off toinfinity

Same observation

Sometimes orbit of 0 goes to infinity, other times it does not.

The Mandelbrot Set:

All c-values for which orbit of 0 does NOT go to infinity.

Why do we care about the orbit of 0?

The Mandelbrot Set:

All c-values for which orbit of 0 does NOT go to infinity.

As we shall see, the orbit of the critical point determines just about everything for z2 + c.

0 is the critical point of z2 + c.

Algorithm for computing M

Start with a grid of complex numbers


Each grid point is a complex c-value.


Compute the orbitof 0 for each c. Ifthe orbit of 0 escapes,color that grid point.

red = fastest escape



orange = slower



yellowgreenblueviolet


Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.


Compute the orbitof 0 for each c. Ifthe orbit of 0 does not escape, leave that grid pointblack.

The eventual orbit of 0



3-cycle


3-cycle


3-cycle


3-cycle


3-cycle


3-cycle


3-cycle


3-cycle


3-cycle




4-cycle


4-cycle


4-cycle


4-cycle


4-cycle


4-cycle


4-cycle


4-cycle




5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


5-cycle


2-cycle


2-cycle


2-cycle


2-cycle


2-cycle


fixed point


fixed point


fixed point


fixed point


fixed point


fixed point


fixed point


fixed point


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


goes to infinity


gone to infinity

One reason for the importance of the critical orbit:

If there is an attracting cycle for z2 + c,then the orbit of 0 must tend to it.

How understand the of the bulbs?periods

How understand the of the bulbs?periods

junction point

three spokes attached

Period 3 bulb

junction point

three spokes attached

Period 4 bulb

Period 5 bulb

Period 7 bulb

Period 13 bulb

Filled Julia Set:

Filled Julia Set:

Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.

Example: z2

Seed:

0

In filled Julia set?

Example: z2

Seed:

0 Yes


Example: z2

Seed:

0 Yes

1


Example: z2

Seed:

0 Yes

1 Yes


Example: z2

Seed:

0 Yes

1 Yes

-1


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i Yes


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i Yes

2i


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i Yes

2i No


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i Yes

2i No

5


Example: z2

Seed:

0 Yes

1 Yes

-1 Yes

i Yes

2i No

5 No way


Filled Julia Set for z 2



All seeds on and inside the unit circle.

i

1-1

The Julia Set is the boundary of the filled Julia set



That’s where the map is “chaotic”




That’s where the map is “chaotic”Nearby orbits behave very differently





































Other filled Julia sets


c = 0






c = -1




c = -1






c = -1






c = -1






c = -1






c = -1






c = -1






c = -1






c = -.12+.75i




c = -.12+.75i










c = -.12+.75i






c = -.12+.75i






c = -.12+.75i




c = -.12+.75i





If c is in the Mandelbrot set, then the filled Julia set is always a connected set.










But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.




c = .3






c = .3






c = .3






c = .3






c = .3






c = -.8+.4i



Another reason why we use the orbit ofthe critical point to plot the M-set:

Theorem: (Fatou & Julia) For z2 + c:



If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.



But if the orbit of 0 does not go to infinity,the Julia set is connected (just one piece).

If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

Animations:

In and out of M

arrangementof the bulbs

Saddle node

Period doubling

Period 4 bifurcation

http://explosion-2.qt.mov/

http://explosion-4.qt.mov/

How do we understand the arrangement of the bulbs?



How do we understand the arrangement of the bulbs?



Assign a fraction p/q to eachbulb hanging off the main cardioid.

q = period of the bulb





Where is the smallest spoke in relationto the “principal spoke”?

p/3 bulb

principal spoke



1/3 bulb



principal spoke

The smallest spoke is located 1/3 ofa turn in the counterclockwise direction

from the principal spoke.



1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





1/3 bulb

1/3





??? bulb

1/3





1/4 bulb

1/3





1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



1/4 bulb

1/3



1/4



??? bulb

1/3

1/4





2/5 bulb

1/3

1/4





2/5 bulb

1/3

1/4



2/5



2/5 bulb

1/3

1/4



2/5



2/5 bulb

1/3

1/4



2/5



2/5 bulb

1/3

1/4



2/5



2/5 bulb

1/3

1/4



2/5



??? bulb

1/3

1/4



2/5



3/7 bulb

1/3

1/4



2/5



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



2/5

3/7



3/7 bulb

1/3

1/4



3/7

2/5



??? bulb

1/3

1/43/7

2/5





1/2 bulb

1/3

1/43/7



1/2

2/5



1/2 bulb

1/3

1/43/7



1/2

2/5



1/2 bulb

1/3

1/43/7



1/2

2/5



1/2 bulb

1/3

1/43/7



1/2

2/5



??? bulb

1/3

1/43/7

1/2

2/5





2/3 bulb

1/3

1/43/7

1/2 QuickTime™ and aTIFF (LZW) decompressor


2/3

2/5



2/3 bulb

1/3

1/43/7



2/3

2/5



2/3 bulb

1/3

1/43/7



2/3

2/5



2/3 bulb

1/3

1/43/7



2/3

2/5



2/3 bulb

1/3

1/43/7



2/3

2/5



2/3 bulb

1/3

1/43/7



2/3

2/5



How to count



1/4

How to count



1/3

1/4

How to count



1/3

1/42/5

How to count



1/3

1/42/5

3/7

How to count



1/3

1/42/5

3/7

1/2

How to count



1/3

1/42/5

3/7

1/2

2/3

How to count



1/3

1/42/5

3/7

1/2

2/3

The bulbs are arranged in the exactorder of the rational numbers.

How to count



1/3

1/42/5

3/7

1/2

2/3

The bulbs are arranged in the exactorder of the rational numbers.

1/101

32,123/96,787

How to count

Animations:

Mandelbulbs

Spiralling fingers

How to add

How to add

1/2

How to add

1/2

1/3

How to add

1/2

1/3

2/5

How to add

1/2

1/3

2/5

3/7

+ =

1/2 + 1/3 = 2/5

+ =

1/2 + 2/5 = 3/7

221/2

0/1

Here’s an interesting sequence:

221/2

0/1

Watch the denominators

1/3

221/2

0/1


1/3

2/5

221/2

0/1


1/3

2/5

3/8

221/2

0/1


1/3

2/5

3/85/13

221/2

0/1

What’s next?

1/3

2/5

3/85/13

221/2

0/1

What’s next?

1/3

2/5

3/85/13

8/21

221/2

0/1

The Fibonacci sequence

1/3

2/5

3/85/13

8/2113/34

The Farey Tree

€

0

1

€

1

1

The Farey Tree

€

0

1

€

1

1

How get the fraction in betweenwith the smallest denominator?

The Farey Tree

€

0

1

€

1

1

€

1

2

How get the fraction in betweenwith the smallest denominator?

Farey addition

The Farey Tree

€

0

1

€

1

1

€

1

2

€

1

3

€

2

3

The Farey Tree

€

0

1

€

1

1

€

1

2

€

1

3

€

2

3

€

2

5

€

1

4

€

3

5

€

3

4

The Farey Tree

€

0

1

€

1

1

€

1

2

€

1

3

€

2

3

€

2

5

€

1

4

€

3

5

€

3

4

€

3

8

€

5

13

....

essentially the golden number

Another sequence (denominatorsonly)

1

2


1

2

3


1

2

3

4


1

2

3

4

5


1

2

3

4

5

6


1

2

3

4

5

67

sequence

1

2

3

4

5

67

Devaney

The Dynamical Systems and Technology Project at Boston University

website: math.bu.edu/DYSYS:

Have fun!

Mandelbrot set explorer;Applets for investigating M-set;Applets for other complex functions;Chaos games, orbit diagrams, etc.

Farey.qt

Farey tree

D-sequence

Continued fraction expansion

Far from rationals

Other topics

Website


Let’s rewrite the sequence:

1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, ..... as a continued fraction:


12

= 12

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


13

= 12 + 1

1

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


25

= 12 + 1

1 + 11

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


38

= 12 + 1

1 + 11 1

1+

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


= 12 + 1

1 + 11 1

1+

11

+

513

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


= 12 + 1

1 + 11 1

1+

11

+

821

11

+

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


= 12 + 1

1 + 11 1

1+

11

+

1334

11

+

11

+

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....


= 12 + 1

1 + 11 1

1+

11

+

1334

11

+

11

+

essentially the1/golden number

the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

We understand what happens for

= 1a + 1

b + 1c 1

d+

1e

+

1f

+

1g

+

where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!!

etc.€

θ

The real way to prove all this:

Need to measure: the size of bulbs the length of spokes the size of the “ears.”

There is an external Riemann map : C - D C - Mtaking the exterior of the unit disk to the exterior of the Mandelbrot set.



€

Φ

€

Φ



€

Φ

€

Φ takes straight rays in C - D to the “external rays” in C - M

01/2

1/3

2/3 €

γ0

€

γ1/3

€

γ2/3€

γ1/2

external ray of angle 1/3

€

1

3→2

3→1

3→

1

7→2

7→4

7→1

7→

1

5→2

5→4

5→3

5→1

5→

Suppose p/q is periodic of period k under doubling mod 1:

period 2

period 3

period 4

€

1

3→2

3→1

3→

1

7→2

7→4

7→1

7→

1

5→2

5→4

5→3

5→1

5→

Suppose p/q is periodic of period k under doubling mod 1:

period 2

period 3

period 4

Then the external ray of angle p/qlands at the “root point” of a period k bulb in the Mandelbrot set.



€

Φ0

1/3

2/3

€

γ00 is fixed under angle doubling, so lands at the cusp of the main cardioid.

€

γ0



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

€

γ1/3

€

γ2/31/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb.

2



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

€

γ1/3

€

γ2/3And if lies between 1/3 and 2/3,then lies between and .

2

€

θ

€

γθ

€

θ

€

γθ



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

So the size of the period 2 bulb is, by definition, the length of the set of rays

between the root point rays, i.e., 2/3-1/3=1/3.

2



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

1/15 and 2/15 have period 4, andare smaller than 1/7....

1/72/7

3/7

4/7

5/7

6/7

€

γ1/7

€

γ2/7

€

γ3/7

€

γ4 /7

€

γ5/7

€

γ6/7

2

3

3

1/15

2/15



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

1/15 and 2/15 have period 4, andare smaller than 1/7....

1/72/7

3/7

4/7

5/7

6/7

€

γ1/7

€

γ2/7

€

γ3/7

€

γ4 /7

€

γ5/7

€

γ6/7

2

3

3

1/15

2/15

€

γ1/15€

γ2/15



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0

1/72/7

3/7

4/7

5/7

6/7

€

γ1/7

€

γ2/7

€

γ3/7

€

γ4 /7

€

γ5/7

€

γ6/7

2

3

3

1/15

2/15

€

γ1/15€

γ2/15

3/15 and 4/15 have period 4, andare between 1/7 and 2/7....



€

Φ0

1/3

2/3

€

γ1/3

€

γ2/3

€

γ0


1/72/7

3/7

4/7

5/7

6/7

€

γ1/7

€

γ2/7

€

γ3/7

€

γ4 /7

€

γ5/7

€

γ6/7

2

3

3

1/15

2/15

€

γ1/15€

γ2/15

1/72/7


1/72/7


3/154/15



So what do we know about M?

All rational external rays land at a single point in M.




All rational external rays land at a single point in M.

Rays that are periodic under doubling land at root points of a bulb.

Non-periodic rational raysland at Misiurewicz points(how we measure lengthof antennas).




“Highly irrational” rays also land at unique points, and we understand what goes on here.

“Highly irrational" = “far”from rationals, i.e.,

€

θ −pq>c

qk

So what do we NOT know about M?



But we don't know if irrationals that are “close” to rationals land.

So we won't understandquadratic functions untilwe figure this out.

MLC Conjecture:

The boundary of the M-setis “locally connected” ---if so, all rays land and we are in heaven!. But if not......







The Dynamical Systems and Technology Project at Boston University

website: math.bu.edu/DYSYS

Have fun!

A number is far from the rationals if:

€

θ

€

|θ − p /q |

€

>


€

θ

€

|θ − p /q |

€

>

€

c /qk


€

θ

€

|θ − p /q |

€

>

€

c /qk

This happens if the “continued fraction expansion” of has only bounded terms.

€

θ

Documents

Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z))