Continued Fraction Expansions: Birkhoff Spectrum, Lyapunov ... · Outlines † Introduction † Gauss map and pressures † Birkhoff spectrum for logarithm of partial quotients †

Continued Fraction Expansions:Birkhoff Spectrum,Lyapunov Spectrum

Ling-Min LIAO

(Joint with A. H. FAN, B. W. WANG and J. WU)LAMFA, CNRS UMR 6140, Amiens

Continued Fraction Expansions: Birkhoff Spectrum, Lyapunov Spectrum. May 25th, 2006 Merlimont – p. 1

Outlines• Introduction

• Gauss map and pressures

• Birkhoff spectrum for logarithm of partialquotients

• Lyapunov spectrum


Introduction• -Problems

• -Previous works

• -Our setting

• -Main results


Problems• Dynamical system (X, T )

• Birkhoff average of φ : X → R(C)(provided the limit exists):φ(x) := limn→∞

1n

∑n−1j=0 φ(T jx)

• Lyapunov exponent(provided the limit exists):λ(x) := limn→∞

1n

∑n−1j=0 log |T ′|(T jx)

• Dimension spectra:• Birkhoff spectrum for φ:

f(α) := dimH

{

x ∈ X; φ(x) = α}

,

• Lyapunov spectrum:g(β) := dimH {x ∈ X; λ(x) = β} .

• Entropy spectrum: dimH → htop.




1n



1n

∑n−1j=0 log |T ′|(T jx)


f(α) := dimH

{

x ∈ X; φ(x) = α}

,






1n



1n

∑n−1j=0 log |T ′|(T jx)


f(α) := dimH

{

x ∈ X; φ(x) = α}

,






1n



1n

∑n−1j=0 log |T ′|(T jx)


f(α) := dimH

{

x ∈ X; φ(x) = α}

,




Problems(continued)• Fast Birkhoff average of φ : X → R(C)(provided the limit exists):

φϕ(x) := limn→∞

1

ϕ(n)

n−1∑

j=0

φ(T jx).

• (weak polynomial) ϕ(n) = n log n,• (polynomial) ϕ(n) = nα (α > 1),• (exponential) ϕ(n) = an (a > 1).

• Fast Birkhoff spectrum for φ:f(α) := dimH

{

x ∈ X; φϕ(x) = α}

,


Problems(continued)• Fast Birkhoff average of φ : X → R(C)(provided the limit exists):

φϕ(x) := limn→∞

1

ϕ(n)

n−1∑

j=0

φ(T jx).

• (weak polynomial) ϕ(n) = n log n,• (polynomial) ϕ(n) = nα (α > 1),• (exponential) ϕ(n) = an (a > 1).

• Fast Birkhoff spectrum for φ:f(α) := dimH

{

x ∈ X; φϕ(x) = α}

,


Previous works• Case of symbolic dynamics on finite alphabet.

• A-H. Fan, D-J. Feng and J. Wu (1998,2001):entropy spectra for continuous φ

• E. Olivier (1998, 1999): dimension spectra for continuous φ

• L. Barreira, Y. Pesin and J. Schemeling (1997):Spectra of Gibbs measure corresponding to a Hölder continuous φ

• Case of conformal expanding maps.• Y. Pesin and H. Weiss (1997): Multifractal of Gibbs measure• H. Weiss (1999): Lyapunov spectra

• Case of symbolic dynamics on infinite alphabet• M. Pollicott and H. Weiss (1999):

Continued fraction system, Lyapunov spectrum"concave on a certain interval".

• M. Kesseböhmer and B. Stratmann(preprint)




















• symbolic dynamics on finite alphabet,

• conformal expanding maps.

0 α0 α1 α2 α

1

f(α)

Birkhoff spectrum for continuous or Holder φ


Our setting→ ([0, 1], T ) continued fraction system

• Gauss map T : [0, 1] → [0, 1] :T (0) := 0, T (x) := 1

x(mod 1), for x ∈ (0, 1]

• invariant ergodic measure µG : dµG = 1(1+x) log 2dx.

• continued fraction expansion:x = 1

a1(x) +1

a2(x) +1

a3(x) +. . .

where a1(x) = b1/xc, and an(x) = a1(Tn−1(x)), for n ≥ 2.

→ Aim: To determine(1) Birkhoff spectrum for log a1(x),

(2) Fast Birkhoff spectra for log a1(x),

(3) Lyapunov spectrum (Birkhoff spectrum for log |T ′(x)|).


Main results(Birkhoff spectrum)→ Birkhoff average of φ(x) := log a1(x):

γ(x) := limn→∞

1

n

n−1∑

j=0

log a1(Tj(x)) = lim

n→∞

1

n

n∑

j=1

log aj(x)

• For almost all x ∈ (0, 1),γ(x) = ξ0 :=

∫

log a1(x)dµG = 2.6854...

Theorem 1• (1) γ(x) attains any value in [0,∞)

• (2) spectrum function t(ξ) := dimH{x ∈ [0, 1] : γ(x) = ξ}

is analytic on (0,∞), neither concave nor convex,

• (3) limξ→0 t(ξ) = 0, t(ξ0) = 1 and limξ→∞ t(ξ) = 1/2,

• (4) t(ξ) is increasing on (0, ξ0) and decreasing on (ξ0,∞),

• (5) limξ→0 t′(ξ) = ∞ and limξ→∞ t′(ξ) = 0.


Main results(Birkhoff spectrum)→ Birkhoff average of φ(x) := log a1(x):

γ(x) := limn→∞

1

n

n−1∑

j=0

log a1(Tj(x)) = lim

n→∞

1

n

n∑

j=1

log aj(x)

• For almost all x ∈ (0, 1),γ(x) = ξ0 :=

∫

log a1(x)dµG = 2.6854...

Theorem 1• (1) γ(x) attains any value in [0,∞)

• (2) spectrum function t(ξ) := dimH{x ∈ [0, 1] : γ(x) = ξ}

is analytic on (0,∞), neither concave nor convex,

• (3) limξ→0 t(ξ) = 0, t(ξ0) = 1 and limξ→∞ t(ξ) = 1/2,

• (4) t(ξ) is increasing on (0, ξ0) and decreasing on (ξ0,∞),

• (5) limξ→0 t′(ξ) = ∞ and limξ→∞ t′(ξ) = 0.


Birkhoff spectrum for log a1(x)

0

12

t(ξ)

1

ξ0 = 2.6854 ξ

t(ξ) = dimH{x ∈ [0, 1] : γ(x) = ξ}


Main results(Lyapunov spectrum)→ Lyapunov exponent:

λ(x) := limn→∞

1

nlog |(T n)′(x)| = lim

n→∞

1

n

n−1∑

j=0

log |T ′(T j(x))|

• For almost all x ∈ (0, 1),λ(x) = λ0 :=

∫

log |T ′(x)|dµG = π2

6 log 2 = 2.37314...

Theorem 2• (1) λ(x) attains any value in [γ0,∞),where γ0 = 2 log(

√5+12 ),

• (2) spectrum function g(β) := dimH{x ∈ [0, 1] : λ(x) = β}

is analytic on (γ0,∞), neither concave nor convex,

• (3) limβ→γ0g(β) = 0, g(λ0) = 1 and limβ→∞ g(β) = 1/2,

• (4) g(β) is increasing on (γ0, λ0) and decreasing on (λ0,∞),

• (5) limβ→γ0g′(β) = ∞ and limβ→∞ g′(β) = 0.


Main results(Lyapunov spectrum)→ Lyapunov exponent:

λ(x) := limn→∞

1

nlog |(T n)′(x)| = lim

n→∞

1

n

n−1∑

j=0

log |T ′(T j(x))|

• For almost all x ∈ (0, 1),λ(x) = λ0 :=

∫

log |T ′(x)|dµG = π2

6 log 2 = 2.37314...

Theorem 2• (1) λ(x) attains any value in [γ0,∞),where γ0 = 2 log(

√5+12 ),

• (2) spectrum function g(β) := dimH{x ∈ [0, 1] : λ(x) = β}

is analytic on (γ0,∞), neither concave nor convex,

• (3) limβ→γ0g(β) = 0, g(λ0) = 1 and limβ→∞ g(β) = 1/2,

• (4) g(β) is increasing on (γ0, λ0) and decreasing on (λ0,∞),

• (5) limβ→γ0g′(β) = ∞ and limβ→∞ g′(β) = 0.


Lyapunov spectrum

0

12

g(β)

1

γ0 λ0 = 2.37314 β

g(β) = dimH{x ∈ [0, 1] : λ(x) = β}


Main results(Fast Birkhoff spectra)→ Fast Birkhoff average of φ(x) := log a1(x):

γϕ(x) := limn→∞

1

ϕ(n)

n−1∑

j=0

log a1(Tj(x)) = lim

n→∞

1

ϕ(n)

n∑

j=1

log aj(x)

Theorem 3 Suppose (ϕn − ϕn−1) ↑ ∞ and limn→∞

ϕn+1−ϕn

ϕn−ϕn−1:= a ≥ 0 exists.

Then for all γ > 0, dimH({γϕ(x) = γ}) = 1/(a + 1).

• Special cases:• (weak polynomial) ϕ(n) = n log n, → 1/2;• (polynomial) ϕ(n) = nα (α > 1), → 1/2;• (exponential) ϕ(n) = an (a > 1), → 1/(a + 1).




1

ϕ(n)

n−1∑

j=0

log a1(Tj(x)) = lim

n→∞

1

ϕ(n)

n∑

j=1

log aj(x)


ϕn+1−ϕn







1

ϕ(n)

n−1∑

j=0

log a1(Tj(x)) = lim

n→∞

1

ϕ(n)

n∑

j=1

log aj(x)


ϕn+1−ϕn





Gauss map and pressures• D := {(t, q) : 2t − q > 1}.

• potential φt,q:

φt,q(x) := −t log |T ′(x)| + q log a1(x) (t, q) ∈ D.

• Ruelle operator Lφt,q:

Lφt,qg(x) :=

∑

y∈T−1x

eφt,q(y)g(y) =

∞∑

i=1

a1(1

i+x)q

(i + x)2tg(

1

i + x).

L∗φt,q

: the conjugate operator

• pressure: P (t, q) := limn→∞1n

log sup[0,1] Lnφt,q

1(x)






Lφt,qg(x) :=

∑

y∈T−1x

eφt,q(y)g(y) =

∞∑

i=1

a1(1

i+x)q

(i + x)2tg(

1

i + x).

L∗φt,q




1(x)






Lφt,qg(x) :=

∑

y∈T−1x

eφt,q(y)g(y) =

∞∑

i=1

a1(1

i+x)q

(i + x)2tg(

1

i + x).

L∗φt,q




1(x)






Lφt,qg(x) :=

∑

y∈T−1x

eφt,q(y)g(y) =

∞∑

i=1

a1(1

i+x)q

(i + x)2tg(

1

i + x).

L∗φt,q




1(x)


→ Ruelle Theory and thermodynamic formalism in countable alphabetsymbolic systems (P. Walters, D. Mayer, R. D. Mauldin, M. Urbanski, O.Sarig, O. Jenkinson, C. Liverani, B. Saussol, S. Vaienti, P. Hanus etc.)→ By the work of Hanus, Mauldin and Urbanski(2002), we get:Proposition 1

• (1) P (t, q) is analytic, strictly convex on D

• (2) P (t, q) is strictly decreasing and strictly convex with respect to t. Inother words, ∂P

∂t(t, q) < 0 and ∂2P

∂t2(t, q) > 0.

• (3) P (t, q) is strictly increasing and strictly convex with respect to q. Inother words, ∂P

∂q(t, q) > 0 and ∂2P

∂q2 (t, q) > 0.


Proposition 2 For (t, q) ∈ D, we have

−t log 4 + log ζ(2t − q) ≤ P (t, q) ≤ log ζ(2t − q),

where ζ(·) is the famous Riemann zeta function, defined byζ(s) :=

∑∞n=1

1ns .

Consequently, for any point (t0, q0) on the line 2t − q = 1, we haveP (t, q) → ∞ as (t, q) → (t0, q0).


Birkhoff spectrum for log a1(x)

Theorem 4 The Birkhoff spectrum for log a1(x), t(ξ) = dimH(Eξ) where

Eξ ={

x ∈ (0, 1) : limn→∞log a1(x)+···+log an(x)

n= ξ

}

is a functiondescribed by the following:

0

12

t(ξ)

1

ξ0 ξ


Construction of spectrum function t(ξ)

Recall that ξ0 =∫

log a1(x)µG.Let D0 := {(t, q) : 2t − q > 1, 0 < t ≤ 1}.Proposition 3 For any ξ ∈ (0,∞), the system

P (t, q) = qξ,∂P

∂q(t, q) = ξ

(1)

admits a unique solution (t(ξ), q(ξ)) ∈ D0, and

• (1) For ξ = ξ0, the solution is (t(ξ0), q(ξ0)) = (1, 0).

• (2) The function t(ξ) and q(ξ) are analytic.


Geometric solution of (t(ξ), q(ξ))

q

P (t, q)

t = 1

t = t(ξ)t = 1

2

t = 0

q(ξ)

ξ0

0 1−1

ξ


Properties of t(ξ)

• limξ→0

t(ξ) = 0 and limξ→∞

t(ξ) = 1/2,

• t′(ξ) = q(ξ)∂P∂t

(t(ξ),q(ξ)),

• q(ξ) < 0 if ξ < ξ0, q(ξ) = 0 if ξ = ξ0 and q(ξ) > 0 if ξ > ξ0,

• t′(ξ) > 0 if ξ < ξ0; t′(ξ) = 0 if ξ = ξ0 and t′(ξ) < 0 if ξ > ξ0,

• limξ→0 q(ξ) = −∞, limξ→∞ q(ξ) = 0,

• limξ→0 t′(ξ) = +∞ and limξ→∞ t′(ξ) = 0,

• q′(ξ) =1−t′(ξ) ∂2P

∂t∂q(t(ξ),q(ξ))

∂2P

∂q2 (t(ξ),q(ξ)),

• t′′(ξ) =t′(ξ)2 ∂2P

∂t2(t(ξ),q(ξ))−q′(ξ)2 ∂2P

∂q2 (t(ξ),q(ξ))

− ∂P∂t

(t(ξ),q(ξ)),

• t′′(ξ0) < 0,

• There exists ξ1 > ξ0 such that t′′(ξ1) > 0.


Lyapunov Spectrum• Consider log |T ′(x)| instead of log a1(x)

• φt,q = −t log |T ′| + q log |T ′| = −(t − q) log |T ′|

• D1 := {(t, q); t − q > 1/2}

• limq→−∞∂P∂q

(t, q) = 2 log 1+√

52

• limq→t− 12

∂P∂q

(t, q) = +∞


Lyapunov Spectrum function

0

12

t(β)

1

γ0 λ0 = 2.37314 β

t(β) = dimH{x ∈ [0, 1] : λ(x) = β}


Construction of Lyapunov Spectrum function t(β)

12− 1

2 β

t = 0

t = 12

t(β)

t = 1

0 q

P (t, q)


Thank you for your attention!


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Continued Fraction Expansions: Birkhoff Spectrum, Lyapunov ... · Outlines † Introduction † Gauss map and pressures † Birkhoff spectrum for logarithm of partial quotients †