Orderings of the rationals: dynamical systems and ... · Orderings of the rationals: dynamical systems and statistical mechanics Stefano Isola Università di Camerino ... I Giampieri,

Orderings of the rationals: dynamicalsystems and statistical mechanics

Stefano IsolaUniversità di Camerino

[email protected]

https://unicam.it/∼stefano.isola/index.html

MECCANICABologna, 27-30 August 2008

Number theory, statistical mechanics, dynamical systems,spectral theory

I I., On the spectrum of Farey and Gauss maps, Nonlinearity15 (2002), 1521-1539

I Giampieri, I., A one-parameter family of analytic Markovinterval maps with an intermittency transition, Disc. Cont.Dyn. Sys. 12 (2005), 115-136

I Degli Esposti, I., Knauf, Generalized Farey Trees, TransferOperators and Phase Transitions, Commun. Math. Phys.275 (2007), 297-329

I Bonanno, Graffi, I., Spectral analysis of transfer operatorsassociated to Farey fractions, Atti Accad. Naz. Lincei Cl.Sci. Fis. Mat. Natur. Rend. Lincei 19 (2007), 1-23

I Bonanno, I., Ordering of the rationals and dynamicalsystems, (2008) submitted

I Bonanno, Degli Esposti, I., Knauf, Geodesic and horocycleflows on the modular domain and the Minkowski questionmark, in preparation

Number theory,

statistical mechanics, dynamical systems,spectral theory







Number theory, statistical mechanics,

dynamical systems,spectral theory







Number theory, statistical mechanics, dynamical systems,

spectral theory





















The Stern-Brocot tree

Binary (genealogical) tree which contains each positiverational number exactly once.

Farey sum (procreation):

pq⊕ p′

q′=

p + p′

q + q′

The child pq ⊕

p′q′ always lies between the parents p

q and p′q′ :

pq

<p + p′

q + q′<

p′

q′




pq⊕ p′

q′=

p + p′

q + q′

The child pq ⊕


q and p′q′ :

pq

<p + p′

q + q′<

p′

q′




pq⊕ p′

q′=

p + p′

q + q′

The child pq ⊕


q and p′q′ :

pq

<p + p′

q + q′<

p′

q′




pq⊕ p′

q′=

p + p′

q + q′

The child pq ⊕


q and p′q′ :

pq

<p + p′

q + q′<

p′

q′

Starting from the ancestors 0 and ∞ ‘in lowest terms’ we get:

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CONNECTION TO CONTINUED FRACTIONS

The elements of the S-B tree of depth d are exactly thoserational numbers x ∈ Q+ whose continued fraction expansion

x = a0 +1

a1 + 1

a2 +1

. . . + 1an

≡ [a0; a1, . . . , an]

with a0 ≥ 0, ai ≥ 1 (0 < i < n) and an > 1, satisfies

n∑i=0

ai = d

EXAMPLE: 85 has depth d = 5 and 8

5 = 1 + 1

1 +1

1 +12



x = a0 +1

a1 + 1

a2 +1

. . . + 1an

≡ [a0; a1, . . . , an]


n∑i=0

ai = d


5 = 1 + 1

1 +1

1 +12



x = a0 +1

a1 + 1

a2 +1

. . . + 1an

≡ [a0; a1, . . . , an]


n∑i=0

ai = d


5 = 1 + 1

1 +1

1 +12

THE SLOW CONTINUED FRACTION ALGORITHM

Given x ∈ R+ there is a unique sequence {xk}k≥1 on the tree,which starts with x1 = 1 and converges to x , finite if x ∈ Q+,infinite otherwise:

if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2

then xk = k if k ≤ a0 + 1, and for k > a0 + 1

xk =r pn−1 + pn−2

r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an

=⇒ xk is called a Farey convergent (FC) of x . If r = an thenxk = pn/qn = [a0; a1, a2, . . . , an−1, an] is the usual continuedfraction convergent (CFC) of x .


Given x ∈ R+ there is a unique sequence {xk}k≥1 on the tree,which starts with x1 = 1 and converges to x ,

finite if x ∈ Q+,infinite otherwise:

if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an




if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an




if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an




if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an




if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an




if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an

=⇒ xk is called a Farey convergent (FC) of x .

If r = an thenxk = pn/qn = [a0; a1, a2, . . . , an−1, an] is the usual continuedfraction convergent (CFC) of x .



if x = [a0; a1, a2, . . . ]

andp−2 = 0, p−1 = 1, pn = an pn−1 + pn−2

q−2 = 1, q−1 = 0, qn = an qn−1 + qn−2



r qn−1 + qn−2, k =

n−1∑i=0

ai + r , 1 ≤ r ≤ an


GROWTH OF THE DENOMINATORS

For x ∈ (0, 1) the CFC’s denominators qn typically growexponentially fast:

log qn

n→ π2

12 log 2almost everywhere

Write a FC as xk = tk/sk (note that min{sk} = k andmax{sk} = fk , the k−th Fibonacci number).If k =

∑ni=1 ai + r then qn−1 < sk ≤ qn.

Moreover (Khinchin and Lévy):

1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2





1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2





1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2


Write a FC as xk = tk/sk (note that min{sk} = k andmax{sk} = fk , the k−th Fibonacci number).

If k =∑n

i=1 ai + r then qn−1 < sk ≤ qn.Moreover (Khinchin and Lévy):

1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2





1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2





1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure



log qn

n→ π2





1n log n

n∑i=1

ai →1

log 2in measure

Thereforelog sk

k∼ π2

12 log kin measure

BINARY CODING

To the sequence {xk} one can associate a binary sequenceσ ∈ {0, 1}N obtained by coding the jumps along the path:

0 1

0 1 0 1

0 1 0 1 0 1 0 1

x2

x3

x4

x1

x = [a0; a1, a2, . . . ] ⇐⇒ σ(x) = 1a00a11a2 . . .

BINARY CODING


0 1

0 1 0 1

0 1 0 1 0 1 0 1

x2

x3

x4

x1

x = [a0; a1, a2, . . . ] ⇐⇒ σ(x) = 1a00a11a2 . . .

BINARY CODING


0 1

0 1 0 1

0 1 0 1 0 1 0 1

x2

x3

x4

x1

x = [a0; a1, a2, . . . ]

⇐⇒ σ(x) = 1a00a11a2 . . .

BINARY CODING


0 1

0 1 0 1

0 1 0 1 0 1 0 1

x2

x3

x4

x1

x = [a0; a1, a2, . . . ] ⇐⇒ σ(x) = 1a00a11a2 . . .

Viceversa, given a (finite or infinite) path σ on the tree, we canreconstruct each element xk as a matrix product:

for z ∈ C

and X =

(a bc d

)∈ SL(2, Z) set X (z) := (az + b)/(cz + d).

With the identification X ≡ X (1) ∈ Q+ we have

x1 ≡(

1 00 1

)and xk ≡

∏1≤i<k

Xi (k ≥ 2)

where

Xi =

{A , if σi = 0B , if σi = 1

A =

(1 01 1

)e B =

(1 10 1

)

Viceversa, given a (finite or infinite) path σ on the tree, we canreconstruct each element xk as a matrix product: for z ∈ C

and X =

(a bc d

)∈ SL(2, Z) set X (z) := (az + b)/(cz + d).


x1 ≡(

1 00 1

)and xk ≡

∏1≤i<k

Xi (k ≥ 2)

where

Xi =

{A , if σi = 0B , if σi = 1

A =

(1 01 1

)e B =

(1 10 1

)


and X =

(a bc d

)∈ SL(2, Z) set X (z) := (az + b)/(cz + d).


x1 ≡(

1 00 1

)and xk ≡

∏1≤i<k

Xi (k ≥ 2)

where

Xi =

{A , if σi = 0B , if σi = 1

A =

(1 01 1

)e B =

(1 10 1

)


and X =

(a bc d

)∈ SL(2, Z) set X (z) := (az + b)/(cz + d).


x1 ≡(

1 00 1

)and xk ≡

∏1≤i<k

Xi (k ≥ 2)

where

Xi =

{A , if σi = 0B , if σi = 1

A =

(1 01 1

)e B =

(1 10 1

)


and X =

(a bc d

)∈ SL(2, Z) set X (z) := (az + b)/(cz + d).


x1 ≡(

1 00 1

)and xk ≡

∏1≤i<k

Xi (k ≥ 2)

where

Xi =

{A , if σi = 0B , if σi = 1

A =

(1 01 1

)e B =

(1 10 1

)

THE FAREY TESSELLATION

H = {z ∈ C : z = x + iy , y > 0} with metric ds2 = dx2+dy2

y2 .

The isometries z 7→ zz+1 and z 7→ z + 1 associated to A e B

generate a tessellation of H starting from the geodesic triangle

G = {z ∈ H |0 < Re z < 1, |z − 12| > 1

2}



y2 .



G = {z ∈ H |0 < Re z < 1, |z − 12| > 1

2}



y2 .



G = {z ∈ H |0 < Re z < 1, |z − 12| > 1

2}



y2 .



G = {z ∈ H |0 < Re z < 1, |z − 12| > 1

2}

finite paths =⇒ geodesics ending at rational points

infinite paths =⇒ geodesics (u, w) with u, w ∈ R \Q

THE PERMUTED S-B TREE

Let∏d

i=1 Xi ∈ SL(2, Z) represent a number x ∈ Q+ of depth d ,and let x̂ ∈ Q+ be given by the reversed product

∏d−1i=0 Xd−i .

The map x 7→ x̂ yields the permuted S-B tree:

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Let∏d

i=1 Xi ∈ SL(2, Z) represent a number x ∈ Q+ of depth d ,

and let x̂ ∈ Q+ be given by the reversed product∏d−1

i=0 Xd−i .The map x 7→ x̂ yields the permuted S-B tree:

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Let∏d

i=1 Xi ∈ SL(2, Z) represent a number x ∈ Q+ of depth d ,and

let x̂ ∈ Q+ be given by the reversed product∏d−1

i=0 Xd−i .The map x 7→ x̂ yields the permuted S-B tree:

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Let∏d


∏d−1i=0 Xd−i .


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Let∏d


∏d−1i=0 Xd−i .


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Let∏d


∏d−1i=0 Xd−i .


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Let∏d


∏d−1i=0 Xd−i .


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SOME (UNEXPECTED) RULES

I Reading the permuted S-B tree row by row, and each rowfrom left to right, the i-th element is given by ξ1 = 1

1 andξi = b(i − 1)/b(i), i > 1, where b(n) is the number ofhyperbinary representations of n (Calkin and Wilf, 2000),e.g. 8 = 23 = 22 + 22 = 22 + 2 + 2 = 22 + 2 + 1 + 1 ⇒b(8) = 4

I We have the recursion ξi+1 = 11−{ξi}+[ξi ]

(Newmann, 2003)

I Reading the permuted tree genealogically starting form 11 ,

under each vertex pq there is the set of descendants{

A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}



1 andξi = b(i − 1)/b(i), i > 1,

where b(n) is the number ofhyperbinary representations of n (Calkin and Wilf, 2000),e.g. 8 = 23 = 22 + 22 = 22 + 2 + 2 = 22 + 2 + 1 + 1 ⇒b(8) = 4


(Newmann, 2003)



A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}



1 andξi = b(i − 1)/b(i), i > 1, where b(n) is the number ofhyperbinary representations of n (Calkin and Wilf, 2000),

e.g. 8 = 23 = 22 + 22 = 22 + 2 + 2 = 22 + 2 + 1 + 1 ⇒b(8) = 4


(Newmann, 2003)



A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}





(Newmann, 2003)



A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}





(Newmann, 2003)



A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}





(Newmann, 2003)



A(

pq

), B(

pq

)}≡{

pp + q

,p + q

q

}

Statistical mechanics

1. RANDOM WALKS

Let Z1,Z2, . . . be a sequence of r.v. on Q+ defined recursivelyin the following way: set Z1 = 1 and if Zk = p

q then eitherZk+1 = p

p+q or Zk+1 = p+qq , both with probability 1

2 .

The sequence (Zk )k≥1 can be regarded as a symmetricrandom walk on the permuted S-B tree.

THEOREM (Bonanno, I., 2008) The random walk (Zk )k≥1 entersany non-empty open interval I = (a, b) ⊂ R+ almost surely.

A more quantitative result in a minute ...


1. RANDOM WALKS




2 .





1. RANDOM WALKS

Let Z1,Z2, . . . be a sequence of r.v. on Q+ defined recursivelyin the following way:

set Z1 = 1 and if Zk = pq then either

Zk+1 = pp+q or Zk+1 = p+q

q , both with probability 12 .





1. RANDOM WALKS




2 .





1. RANDOM WALKS




2 .





1. RANDOM WALKS




2 .





1. RANDOM WALKS




2 .




2. KNAUF’ SPIN CHAIN

To each of the 2d elements of depth d + 1 in the permuted S-Btree one can attach a spin configuration corresponding to itsaddress σ(a/b) ∈ {0, 1}d and the energy Ed(σ) = log b.The ground state is σ(d/1) = 11 · · ·1 with energy log 1 = 0,the most excited state has energy log fd+1.The (canonical) partition function is

Zd(β) =∑

depth( ab )=d+1

b−β

The resulting interaction is ferromagnetic (Knauf, 1993):

jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d


To each of the 2d elements of depth d + 1 in the permuted S-Btree one can attach a spin configuration corresponding to itsaddress σ(a/b) ∈ {0, 1}d and the energy Ed(σ) = log b.

The ground state is σ(d/1) = 11 · · ·1 with energy log 1 = 0,the most excited state has energy log fd+1.The (canonical) partition function is

Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d


To each of the 2d elements of depth d + 1 in the permuted S-Btree one can attach a spin configuration corresponding to itsaddress σ(a/b) ∈ {0, 1}d and the energy Ed(σ) = log b.The ground state is σ(d/1) = 11 · · ·1 with energy log 1 = 0,

the most excited state has energy log fd+1.The (canonical) partition function is

Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d


To each of the 2d elements of depth d + 1 in the permuted S-Btree one can attach a spin configuration corresponding to itsaddress σ(a/b) ∈ {0, 1}d and the energy Ed(σ) = log b.The ground state is σ(d/1) = 11 · · ·1 with energy log 1 = 0,the most excited state has energy log fd+1.

The (canonical) partition function is

Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d



Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d



Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d



Zd(β) =∑

depth( ab )=d+1

b−β


jd(τ) := − 12d

∑σ∈{0,1}d

Ed(σ) · (−1)σ·τ ≥ 0 , τ ∈ {0, 1}d \ 0d

PHASE TRANSITION OF SECOND ORDER AT β = 2

limd→∞

Zd(β) =ζ(β − 1)

ζ(β), Reβ > 2 (Knauf , 1993)

Zd(2) ∼ d2 log d

, d →∞ (I., 2005)

The free energy −β f (β) is real analytic for β < 2 and

−β f (β) ∼ 2− β

− log (2− β)β → 2− (Prellberg, 1992)

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2


limd→∞

Zd(β) =ζ(β − 1)

ζ(β), Reβ > 2 (Knauf , 1993)

Zd(2) ∼ d2 log d

, d →∞ (I., 2005)


−β f (β) ∼ 2− β

− log (2− β)β → 2− (Prellberg, 1992)

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2


limd→∞

Zd(β) =ζ(β − 1)

ζ(β), Reβ > 2 (Knauf , 1993)

Zd(2) ∼ d2 log d

, d →∞ (I., 2005)


−β f (β) ∼ 2− β

− log (2− β)β → 2− (Prellberg, 1992)

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2


limd→∞

Zd(β) =ζ(β − 1)

ζ(β), Reβ > 2 (Knauf , 1993)

Zd(2) ∼ d2 log d

, d →∞ (I., 2005)


−β f (β) ∼ 2− β

− log (2− β)β → 2− (Prellberg, 1992)

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2


limd→∞

Zd(β) =ζ(β − 1)

ζ(β), Reβ > 2 (Knauf , 1993)

Zd(2) ∼ d2 log d

, d →∞ (I., 2005)


−β f (β) ∼ 2− β

− log (2− β)β → 2− (Prellberg, 1992)

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2

3. Generalized S-B trees and spin chains

In the Farey sum assign "more weight to older parents": giventwo neighbours p

q , p′q′ , of depth d − k (1 ≤ k ≤ d) and d resp.,

setp′′

q′′=

p′ + wkpq′ + wkq

, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′

q′and p′q − pq′ = wd−k

and the resulting elements are ratios of polynomials in Z[w ]:

01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2


In the Farey sum assign "more weight to older parents":

giventwo neighbours p


setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2




setp′′

q′′=


, w ∈ [1, 2]

thenpq

<p′′

q′′<

p′



01

ww−1

11

12

w+1w

12+w

1+w2+w

1+2w2w

1+w+w2

w2

ENERGIES OF SPIN CHAINS

100 200 300 400 500

1

2

3

4

5

6

7w=2�

100 200 300 400 500

1

2

3

4

5

6

7w=1.8

100 200 300 400 500

1

2

3

4

5

6

7w=1.6�

100 200 300 400 500

1

2

3

4

5

6

7w=1.4�

100 200 300 400 500

1

2

3

4

5

6

7w=1.2�

100 200 300 400 500

1

2

3

4

5

6

7w=1�

ENERGIES OF SPIN CHAINS

100 200 300 400 500

1

2

3

4

5

6

7w=2�

100 200 300 400 500

1

2

3

4

5

6

7w=1.8

100 200 300 400 500

1

2

3

4

5

6

7w=1.6�

100 200 300 400 500

1

2

3

4

5

6

7w=1.4�

100 200 300 400 500

1

2

3

4

5

6

7w=1.2�

100 200 300 400 500

1

2

3

4

5

6

7w=1�

THEOREM (Degli Esposti, I., Knauf, 2007)

I The interaction of the spin chain is ferromagnetic for eachw ∈ [1, 2].

I There is a monotonically decreasing function βcr (w), withβcr (1) = 2 and βcr (2) = 1, such that the partition fct Zd(β)has a finite limit as d →∞ whenever β > βcr .

I For w = 1 the phase transition at βcr = 2 is of secondorder, but for 1 < w ≤ 2 the first derivative of −β f (β) isdiscontinuous at βcr (first order transition). Moreover, for allw ∈ [1, 2] the magnetization jumps at βcr from 1 to 0.













EXAMPLE: w = 2

Zd(β) =2β − 1− 2d(1−β)

2β − 2

limd→∞

Zd(β) =2β − 12β − 2

, Re β > βcr = 1

−β f (β) =

{(1− β) log 2 , if β < 1

0 , if β ≥ 1

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2

EXAMPLE: w = 2

Zd(β) =2β − 1− 2d(1−β)

2β − 2

limd→∞

Zd(β) =2β − 12β − 2

, Re β > βcr = 1

−β f (β) =

{(1− β) log 2 , if β < 1

0 , if β ≥ 1

-2 -1 1 2 3

-1

-0.5

0.5

1

1.5

2

Some of the above results can be proved by writing thepartition fct, or more generally, sums like

Z (m)d (β) :=

∑depth( a

b )=d+1

e2πi m ab

bβ, m ∈ Z

in the form

Z (m)d (β) =

12

(1 +

d∑k=0

w−kβ/2 P+β

ke2πi m x |x=1

)

where P+β is the transfer operator associated to a map by

which the tree can be dynamically generated, and then usingspectral theory.


Z (m)d (β) :=

∑depth( a

b )=d+1

e2πi m ab

bβ, m ∈ Z

in the form

Z (m)d (β) =

12

(1 +

d∑k=0

w−kβ/2 P+β

ke2πi m x |x=1

)




Z (m)d (β) :=

∑depth( a

b )=d+1

e2πi m ab

bβ, m ∈ Z

in the form

Z (m)d (β) =

12

(1 +

d∑k=0

w−kβ/2 P+β

ke2πi m x |x=1

)




Z (m)d (β) :=

∑depth( a

b )=d+1

e2πi m ab

bβ, m ∈ Z

in the form

Z (m)d (β) =

12

(1 +

d∑k=0

w−kβ/2 P+β

ke2πi m x |x=1

)


which the tree can be dynamically generated,

and then usingspectral theory.


Z (m)d (β) :=

∑depth( a

b )=d+1

e2πi m ab

bβ, m ∈ Z

in the form

Z (m)d (β) =

12

(1 +

d∑k=0

w−kβ/2 P+β

ke2πi m x |x=1

)



Dynamics

Set Y := R+ ∪ {∞} and let S : Y → Y be the (invertible) map

S : x 7→ 11− {x}+ [x ]

=⇒ The forward S-orbit of 11 yields the permuted S-B tree row

by row with increasing depth.

Let moreover F : Y → Y be the (non-invertible) map

F : x 7→

x

1− xif 0 ≤ x < 1

x − 1 if x ≥ 1

=⇒ The permuted S-B tree can be constructed genealogicallyfrom the root 1

1 by writing under each leaf x its descendantsF−1(x).

Dynamics


S : x 7→ 11− {x}+ [x ]




F : x 7→

x

1− xif 0 ≤ x < 1

x − 1 if x ≥ 1



Dynamics


S : x 7→ 11− {x}+ [x ]




F : x 7→

x

1− xif 0 ≤ x < 1

x − 1 if x ≥ 1



Dynamics


S : x 7→ 11− {x}+ [x ]




F : x 7→

x

1− xif 0 ≤ x < 1

x − 1 if x ≥ 1



Dynamics


S : x 7→ 11− {x}+ [x ]




F : x 7→

x

1− xif 0 ≤ x < 1

x − 1 if x ≥ 1



We plot the interval maps S̃ := φ ◦ S ◦ φ−1 and F̃ := φ ◦ F ◦ φ−1

where φ(x) := x/(1 + x) (maps the S-B tree to the Farey tree):

S̃(x) =1

2−{

x1−x

}+[

x1−x

] , F̃ =

x

1− xif 0 ≤ x < 1

2

2− 1x if 1

2 ≤ x ≤ 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:

We plot the interval maps S̃ := φ ◦ S ◦ φ−1 and F̃ := φ ◦ F ◦ φ−1

where φ(x) := x/(1 + x) (maps the S-B tree to the Farey tree):

S̃(x) =1

2−{

x1−x

}+[

x1−x

] , F̃ =

x

1− xif 0 ≤ x < 1

2

2− 1x if 1

2 ≤ x ≤ 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:

CONJUGATIONS AND ERGODIC PROPERTIES

Given x ∈ R+ with c.f.e. x = [a0; a1, a2, . . . ] one may ask whatis the number ρ(x) obtained by interpreting its symbolicsequence σ(x) on the S-B tree as the binary expansion of areal number in (0, 1), i.e.

ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·

We have ρ(x) =? ◦ φ (x) where ? : [0, 1] → [0, 1] is theMinkowski question mark function:

I ?(x) is strictly increasing from 0 to 1 and Hölder continuousof order β = log 2/(

√5 + 1)

I x is rational iff ?(x) is dyadic rationalI x is a quadratic irrational iff ?(x) is a (non-dyadic) rationalI d?(x) vanishes Lebesgue-almost everywhere



ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)




ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)




ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)




ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)




ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)

I x is rational iff ?(x) is dyadic rational

I x is a quadratic irrational iff ?(x) is a (non-dyadic) rationalI d?(x) vanishes Lebesgue-almost everywhere



ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)

I x is rational iff ?(x) is dyadic rationalI x is a quadratic irrational iff ?(x) is a (non-dyadic) rational

I d?(x) vanishes Lebesgue-almost everywhere



ρ(x) = 0 . 11 . . . 1︸︷︷︸a0

00 . . . 0︸︷︷︸a1

11 . . . 1︸︷︷︸a2

· · ·



√5 + 1)


0.2 0.4 0.6 0.8 1x

0.2

0.4

0.6

0.8

1

?�x�

THEOREM (Bonanno, I., 2008) We have the followingconjugations

S = ρ−1 ◦ T ◦ ρ , F = ρ−1 ◦ D ◦ ρ

where T : [0, 1] → [0, 1] is the Von Neumann-Kakutani map(or dyadic rotation):

T (x) := x +32n − 1 , 1− 1

2n−1 ≤ x < 1− 12n , n ≥ 1

and D : [0, 1] → [0, 1] is the doubling map: D(x) := 2x mod 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:


S = ρ−1 ◦ T ◦ ρ , F = ρ−1 ◦ D ◦ ρ


T (x) := x +32n − 1 , 1− 1

2n−1 ≤ x < 1− 12n , n ≥ 1


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:


S = ρ−1 ◦ T ◦ ρ , F = ρ−1 ◦ D ◦ ρ


T (x) := x +32n − 1 , 1− 1

2n−1 ≤ x < 1− 12n , n ≥ 1


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:


S = ρ−1 ◦ T ◦ ρ , F = ρ−1 ◦ D ◦ ρ


T (x) := x +32n − 1 , 1− 1

2n−1 ≤ x < 1− 12n , n ≥ 1


0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Figure:

Facts:

I both T and D preserve the Lebesgue measureI T is strictly ergodic, has rank one and discrete spectrumI T ◦ D = D ◦ T 2

Consequences:

I S preserves the singular measure dρ, is strictly ergodic,has rank one and discrete spectrum

I F preserves several measures. Among them: the measuredρ, which is the measure of maximal entropy (withentropy log 2), and the infinite a.c. measure dx/x (withzero entropy)

I the random walk (Zk )k≥1 enters the open intervalI = (a, b) ⊂ R+ with asymptotic frequency ρ(I)

I S ◦ F = F ◦ S2

Facts:

I both T and D preserve the Lebesgue measure

I T is strictly ergodic, has rank one and discrete spectrumI T ◦ D = D ◦ T 2

Consequences:




I S ◦ F = F ◦ S2

Facts:

I both T and D preserve the Lebesgue measureI T is strictly ergodic, has rank one and discrete spectrum

I T ◦ D = D ◦ T 2

Consequences:




I S ◦ F = F ◦ S2

Facts:


Consequences:




I S ◦ F = F ◦ S2

Facts:


Consequences:




I S ◦ F = F ◦ S2

Facts:


Consequences:


I F preserves several measures.

Among them: the measuredρ, which is the measure of maximal entropy (withentropy log 2), and the infinite a.c. measure dx/x (withzero entropy)


I S ◦ F = F ◦ S2

Facts:


Consequences:


I F preserves several measures. Among them: the measuredρ, which is the measure of maximal entropy (withentropy log 2),

and the infinite a.c. measure dx/x (withzero entropy)


I S ◦ F = F ◦ S2

Facts:


Consequences:




I S ◦ F = F ◦ S2

Facts:


Consequences:




I S ◦ F = F ◦ S2

Facts:


Consequences:




I S ◦ F = F ◦ S2

F AND S AS POINCARÉ MAPS

Considering geodesics (u, w) with u < 0 < w , the action of Fon w can be regarded as a (factor of) the Poincaré mapassociated to the geodesic flow gt : TM → TM on the modularsurface M = SL(2, Z) \H (not compact but of finite hyperbolicarea and gaussian curvature K = −1).





The action of S can be regarded as a Poincaré map associatedto the horocycle flow ht : TM → TM, with a return time whichdepends on the initial point on the section ... joint work inprogress with Bonanno, Degli Esposti and Knauf

The action of S can be regarded as a Poincaré map associatedto the horocycle flow ht : TM → TM, with a return time whichdepends on the initial point on the section

... joint work inprogress with Bonanno, Degli Esposti and Knauf

The action of S can be regarded as a Poincaré map associatedto the horocycle flow ht : TM → TM, with a return time whichdepends on the initial point on the section ... joint work inprogress with Bonanno, Degli Esposti and Knauf

Transfer operators and hyperbolic laplacian

To the inverse branches of F we associate the generalizedtransfer operator (β ∈ C)

(Lβf )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)

In particular L ≡ L2 is called Perron-Frobenius operator andsatisfies ∫

g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx

A function h is the density of an a.c. F -invariant measure iffLh = h. In this case we have h(x) = 1/x . More generally, theeigenvalue eq. Lβf = λf reads

λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1

)and, if λ = 1 this is the Lewis-Zagier functional equation,whose solutions are called period functions.

Transfer operators and hyperbolic laplacianTo the inverse branches of F we associate the generalizedtransfer operator (β ∈ C)

(Lβf )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1



(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2

= f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1



(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1



(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1



(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx

A function h is the density of an a.c. F -invariant measure iffLh = h. In this case we have h(x) = 1/x .

More generally, theeigenvalue eq. Lβf = λf reads

λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1



(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx

A function h is the density of an a.c. F -invariant measure iffLh = h. In this case we have h(x) = 1/x . More generally, theeigenvalue eq. Lβ f = λf reads

λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1

)

and, if λ = 1 this is the Lewis-Zagier functional equation,whose solutions are called period functions.


(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1

)and, if λ = 1 this is the Lewis-Zagier functional equation,

whose solutions are called period functions.


(Lβ f )(x) =∑

y∈F−1(x)

f (y)

|F ′(y)|β/2 = f (x + 1)+

(1

x + 1

)β

f(

xx + 1

)


g ◦ F (x) h(x)dx =

∫g(x)(Lh)(x)dx


λ f (x)− f (x + 1) =1

(x + 1)βf(

xx + 1


A Maass wave form of parameter s ∈ C is a SL(2, Z)-invariantfct u : H → C, vanishing for y →∞ and satisfying∆u = s(1− s)u where ∆ = y2(∂2

x + ∂2y ) is the hyperbolic

Laplacian on H.

A Maass wave form of parameter s ∈ C is a SL(2, Z)-invariantfct u : H → C, vanishing for y →∞ and satisfying∆u = s(1− s)u where ∆ = y2(∂2

x + ∂2y ) is the hyperbolic

Laplacian on H.

There is a one-to-one correspondence between the space ofMaass wave forms u of parameter s ∈ C with Res = 1/2 andthe space of real analytic solutions on R+ of the Lewis-Zagiereq. Lβf = f with β = 2s, which satisfy

f (x) = o(1/x) (x → 0), h(x) = o(1) (x →∞)

(Lewis, Zagier, Ann. Math. 153 (2001), 191-258)

What about the rest of the spectrum of Lβ?

There is a one-to-one correspondence between the space ofMaass wave forms u of parameter s ∈ C with Res = 1/2 andthe space of real analytic solutions on R+ of the Lewis-Zagiereq. Lβf = f with β = 2s, which satisfy

f (x) = o(1/x) (x → 0), h(x) = o(1) (x →∞)

(Lewis, Zagier, Ann. Math. 153 (2001), 191-258)

What about the rest of the spectrum of Lβ?

Alternative formulation

Let P±β be the (signed) transfer operators associated to the

Farey map and acting as

(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1

)]If H(B) denotes the set of fcts holomorphic in B ⊆ C we have

I If f ∈ H({|z − 1| < 1}) then P±β f ∈ H({Rez > 0})

I If P±β f = λf with λ 6= 0 then Jβf = ±f , with

(Jβf )(z) :=1zβ

f(

1z

)Therefore the eigenvalue eqs. P±

β f = λ f are bothequivalent to Lβf = λf

I If u is a Maass form s.t. u(x + iy) = ±u(−x + iy) thenP±

β f = f with β = 2s




(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1

)]

If H(B) denotes the set of fcts holomorphic in B ⊆ C we have



(Jβf )(z) :=1zβ

f(

1z








(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1




(Jβf )(z) :=1zβ

f(

1z








(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1




(Jβf )(z) :=1zβ

f(

1z








(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1




(Jβ f )(z) :=1zβ

f(

1z

)

Therefore the eigenvalue eqs. P±β f = λ f are both

equivalent to Lβf = λfI If u is a Maass form s.t. u(x + iy) = ±u(−x + iy) then

P±β f = f with β = 2s




(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1




(Jβ f )(z) :=1zβ

f(

1z








(P±β f )(x) :=

(1

x + 1

)β [f(

xx + 1

)± f(

1x + 1




(Jβ f )(z) :=1zβ

f(

1z





SOME SPECTRAL PROPERTIES OF P±β

Let Hβ be the Hilbert space of functions f ∈ H({|z − 1| < 1})and such that

f (x) = B [ϕ](x) :=1

xβ

∫ ∞

0e−

tx etϕ(t)mβ(dt)

with ϕ ∈ L2(mβ) and mβ(dt) = tβ−1e−tdt .

THEOREM (I., 2003) P±β B [ϕ] = B [(M + N)ϕ] with

M ϕ(t) = e−tϕ(t) , Nϕ(t) =

∫ ∞

0

Jβ−1

(2√

st)

(st)(β−1)/2 ϕ(s) mβ(ds)

THEOREM (Bonanno, Graffi, I., 2007) For each β ∈ (0,∞) theoperators P±

β : Hβ → Hβ are bounded, self-adjoint andisospectral. Their spectrum is {0} ∪ (0, 1], with (0, 1] purely a.c.


Let Hβ be the Hilbert space of functions f ∈ H({|z − 1| < 1})

and such that

f (x) = B [ϕ](x) :=1

xβ

∫ ∞

0e−

tx etϕ(t)mβ(dt)



M ϕ(t) = e−tϕ(t) , Nϕ(t) =

∫ ∞

0

Jβ−1

(2√

st)

(st)(β−1)/2 ϕ(s) mβ(ds)





f (x) = B [ϕ](x) :=1

xβ

∫ ∞

0e−

tx etϕ(t)mβ(dt)



M ϕ(t) = e−tϕ(t) , Nϕ(t) =

∫ ∞

0

Jβ−1

(2√

st)

(st)(β−1)/2 ϕ(s) mβ(ds)





f (x) = B [ϕ](x) :=1

xβ

∫ ∞

0e−

tx etϕ(t)mβ(dt)



M ϕ(t) = e−tϕ(t) , Nϕ(t) =

∫ ∞

0

Jβ−1

(2√

st)

(st)(β−1)/2 ϕ(s) mβ(ds)





f (x) = B [ϕ](x) :=1

xβ

∫ ∞

0e−

tx etϕ(t)mβ(dt)



M ϕ(t) = e−tϕ(t) , Nϕ(t) =

∫ ∞

0

Jβ−1

(2√

st)

(st)(β−1)/2 ϕ(s) mβ(ds)



Some perspectives

I Generalized eigenfcts of P±β with β ∈ (0,∞)

I Spectrum of P±β with β ∈ C

I Tree expansion and analytic continuation of (generalized)Ruelle and Selberg zeta functions

I Harmonic functions and martingalesI Integrability of spin models

Some perspectives





Some perspectives





Some perspectives





Some perspectives




I Harmonic functions and martingales

I Integrability of spin models

Some perspectives





Some perspectives





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