Entropy spectrum of Lyapunov exponents for typical cocycles

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arXiv:2210.11574v3 [math.DS] 22 Jul 2024

ENTROPY SPECTRUM OF LYAPUNOV EXPONENTS FOR TYPICAL

COCYCLES

REZA MOHAMMADPOUR ID (UPPSALA UNIVERSITY)

reza.mohammadpour@math.uu.se

Abstract. In this paper, we study the size of the level sets of all Lyapunov exponents.

For typical cocycles, we establish a variational relation between the topological entropy

of the level sets of Lyapunov exponents and the topological pressure of the generalized

singular value function.

1. Introduction

Suppose that Xis a compact metric space that is endowed with the metric d. A con-

tinuous map T:X→Xon the compact metric space Xis called a topological dynamical

system (TDS) and we denote it by (X, T ). Let M(X)be the space of all Borel probability

measures on X, and M(X, T )be the space of all T-invariant Borel probability measures

on X.

Assume that f:X→Ris a continuous function. Denote by Snf(x) := Pn−1

k=0 f(Tk(x))

the Birkhoﬀ sum, and we call

lim

n→∞

nSnf(x)

the Birkhoﬀ average.

The Birkhoﬀ average converges to the integral of fwith respect to the ergodic invariant

probability measure µ, almost everywhere. However, there are numerous ergodic invariant

measures where the limit exists but converges to a diﬀerent value. Additionally, there are

plenty of points where the Birkhoﬀ average either does not exist or are not considered

generic points for any ergodic measure. Therefore, we may ask about the size of the set of

points

Ef(α) = x∈X:1

nSnf(x)→αas n→ ∞,

which we call the α-level set of Birkhoﬀ spectrum, for a given value αfrom the set

Lf=α∈R:∃x∈Xand lim

n→∞

nSnf(x) = α,

Date: July 23, 2024.

2010 Mathematics Subject Classiﬁcation. 28A80, 28D20, 37D35, 37H15 .

Key words and phrases. Lyapunov exponents, multifractal formalism, topological entropy, typical

cocycles.

ENTROPY SPECTRUM OF LYAPUNOV EXPONENTS FOR TYPICAL COCYCLES 2

which we call the Birkhoﬀ spectrum. The size is determined using either the Hausdorﬀ

dimension or the topological entropy introduced by Bowen in [12] (See Subsection 2.6).

The topological entropy and Hausdorﬀ spectrum of Birkhoﬀ averages have been intensely

studied by several authors (see, e.g. [8,9,23]) for diﬀerent systems and are well understood

for continuous potentials.

We say that Φ := {log φn}∞

n=1 is a subadditive potential if each φnis a continuous positive-

valued function on Xsuch that

0< φn+m(x)≤φn(x)φm(Tn(x)) ∀x∈X, m, n ∈N.

Moreover, a sequence of continuous functions (potentials) Φ = {log φn}∞

n=1 is said to be

an almost additive potential if there exists a constant C > 0such that for any m, n ∈N,

x∈X, we have

C−1φn(x)φm(Tn)(x)≤φn+m(x)≤Cφn(x)φm(Tn(x)).

By Kingman’s subadditive theorem, for any µ∈ M(X, T )and µalmost every x∈X

such that log+φ1∈L1(µ), the following limit, called the top Lyapunov exponent at x,

exists:

χ(x, Φ) := lim

n→∞

nlog φn(x).

Let A:X→GL(d, R)be a continuous function over a topological dynamical system

(X, T ). We denote the product of Aalong the orbit of xfor time n, where xis an element

of Xand nbelongs to the set of natural numbers, as

An(x) := ATn−1(x)...A(x).

The pair (A, T )is called a matrix cocycle; when the context is clear, we say that Ais a

matrix cocycle. That induces a skew-product dynamics Fon X×Rkby (x, v)7→ X×Rk,

whose n-th iterate is therefore

(x, v)7→ (Tn(x),An(x)v).

If Tis invertible then so is F. Moreover, F−n(x) = (T−n(x),A−n(x)v)for each n≥1,

where

A−n(x) := A(T−n(x))−1A(T−n+1(x))−1...A(T−1(x))−1.

A well-known example of matrix cocycles is one-step cocycles which is deﬁned as follows.

Assume that Σ = {1, ..., k}Zis a symbolic space. Suppose that T: Σ →Σis a shift map,

i.e. T(xl)l∈Z= (xl+1)l∈Z. Given a k-tuple of matrices A= (A1,...,Ak)∈GLd(R)k, we

associate with it the locally constant map A: Σ →GLd(R)given by A(x) = Ax0,that

means the matrix cocycle Adepends only on the zero-th symbol x0of (xl)l∈Z. In this

particular situation, we say that (A, T )is a one-step cocycle; when the context is clear, we

say that Ais a one-step cocycle. For any length nword I=i0,...,in−1,(see Section 2for

the deﬁnition) we denote

AI:= Ain−1. . . Ai0.

Assume that (A1,...,Ak)∈GL(d, R)kgenerates a one-step cocycle A: Σ →GL(d, R).

Motivated by the study of the multifractal formalism of Birkhoﬀ averages, the level set of

the top Lyapunov exponent of certain special subadditive potentials Φ = {log φn}∞

n=1 on

ENTROPY SPECTRUM OF LYAPUNOV EXPONENTS FOR TYPICAL COCYCLES 3

full shifts have been studied in [17,18,13,20], in which φn(x) = kAn(x)k, where k · k

denotes the operator norm. In other words, our focus lies in determining the size of the set

of points

E(α) = x∈Σ : 1

nlog kAn(x)k → αas n→ ∞,

which we call the α-level set of the top Lyapunov exponent, for a given value αfrom the set

L=α∈R:∃x∈Σand lim

n→∞

nlog kAn(x)k=α.

The author [20] and Feng [17] calculated the entropy spectrum of the top Lyapunov

exponent for generic matrix cocycles.

Let µbe an ergodic T-invariant measure. By Oseledets’ theorem, there is a set Y⊂Σ

of full measure such that if x∈Ythen the Lyapunov exponents χ1(x, A)≥χ2(x, A)≥

... ≥χd(x, A), counted with multiplicity, exist. For ~α := (α1,...,αd)∈Rd, we deﬁne the

~α-level set I(~α)of the Lyapunov exponents by

I(~α) = x∈Y:χi(x, A) = αifor i= 1,...,d

(we emphasize that selecting points in Yimplies the assumption of the existence of the

limits.). This suggests to consider, for each ~α := (α1,...,αd)∈Rd, the ~α-level set

E(~α) = x∈Σ : lim

n→∞

nlog σi(An(x)) = αifor i= 1,2,...,d,

where σ1,...,σdare singular values, listed in decreasing order according to multiplicity.

We also deﬁne the Lyapunov spectrum

L=~α ∈Rd:∃x∈Σsuch that lim

n→∞

nlog σi(An(x)) = αifor i= 1,2,...,d.

Note that we have E(~α) = I(~α)(mod0) for every ~α ∈Rd. Therefore, one may also

think of the ~α-level set E(~α)as a level set of the Lyapunov exponents. Our main goal

is to calculate the topological entropy of these sets. More precisely, we want to show

that the topological entropy of the ~α-level set E(~α)is equal to the Legendre transform

of the topological pressure for generic matrix cocycles. Note that a similar statement can

be expressed in probabilistic language to establish a large deviation principle (LDP) for

random matrix products (see [26]).

Feng and Huang [18] calculated the topological entropy of ~α-level sets E(~α)for almost

additive potentials that improves a result of Barreira and Gelfert in [6] on Lyapunov ex-

ponents of nonconformal repellers. Notice that almost additivity condition holds only for

a restrictive family of matrices. For instance, Bárány et al. [3] showed that this condition

for planar matrix tuples is equivalent to domination, which is an open condition but not

generic. One can ﬁnd more information about the multifractal formalism in [7,15,21].

In this paper, we consider typical cocycles that are matrix cocycles with extra assumptions

on some periodic point p∈Σand one of its homoclinic points z∈Σ(see Section 2for the

ENTROPY SPECTRUM OF LYAPUNOV EXPONENTS FOR TYPICAL COCYCLES 4

precise deﬁnition). We call this pair (p, z)a typical pair, and the typicality assumptions

on the pair are suitable generalizations of the proximality and strong irreducibility in the

setting of random product of matrices. Bonatti and Viana [11] introduced the notion of

typical cocycles. They showed that the set of typical cocycles is open and dense in the

set of ﬁber-bunched cocycles and that its complement has inﬁnite codimension. Also, they

proved that typical cocycles have simple Lyapunov exponents with respect to any ergodic

measures with continuous local product structure.

We consider typical cocycles and ~α ∈Rd.Then, we calculate the topological entropy of

the ~α-level set E(~α). For simplicity, we say the level set E(~α)instead of the ~α-level set

E(~α)if there is no confusion about ~α.

We deﬁne Falconer’s singular value function ϕs(A)as follows. Let k∈ {0,...,d−1}

and k≤s < k + 1. Then,

ϕs(A) = σ1(A)···σk(A)σk+1(A)s−k,

and if s≥d, then ϕs(A) = (det(A))s

For q:= (q1,··· , qd)∈Rd, we deﬁne the generalized singular value function ψq1,...,qd(A) :

Rd×d→[0,∞)as

ψq1,...,qd(A) := σ1(A)q1···σd(A)qd= d−1

m=1 

A∧m

qm−qm+1 !



A∧d



qd.

When s∈[0, d], the singular value function ϕs(A(·)) coincides with the generalized

singular value function ψq1,...,qd(A(·)) where

(q1,...,qd) = (1,...,1

|{z }

mtimes

, s −m, 0,...,0),

with m=⌊s⌋. We denote ψq(A) := ψq1,...,qd(A).We should notice that even though

there is some similarity between the previous expressions, Falconer’s singular value function

ϕs(A)is submultiplicative, whereas the generalized singular value function ψq(A)is neither

submultiplicative nor supermultiplicative.

Notice that the limit in deﬁning the topological pressure P(log ψq(A)) exists for any

q∈Rdwhen Ais a typical cocycle (see Section 3).

Theorem A. Assume that (A1,...,Ak)∈GL(d, R)kgenerates a one-step cocycle A: Σ →

GL(d, R).Let A: Σ →GL(d, R)be a typical cocycle. Then

htop(E(~α)) = inf

q∈Rd{P(log ψq(A)) − hq, ~αi}

for all ~α ∈˚

In Section 2 we introduce some notation and preliminaries. In Section 3 we prove the

upper bound of Theorem A. The key idea in the proof of Theorem A is to ﬁnd the dominated

subsystems for typical cocycles, and then we prove that the topological pressure over the

dominated subset converges to the topological pressure over all points that are done in

ENTROPY SPECTRUM OF LYAPUNOV EXPONENTS FOR TYPICAL COCYCLES 5

Section 4. We prove Theorem A for dominated cocycles in Section 5. Finally, we prove

Theorem A in Section 6.

1.1. Acknowledgements. The author thanks Ville Salo, Michal Rams and Kiho Park for

helpful discussions. He also thanks the anonymous referee for their valuable corrections

and suggestions.

2. Preliminaries

2.1. Subshifts of ﬁnite type and standing notations. Assume that Q= (qij )is a

matrix k×kwith qij ∈ {0,1}.The (two sided) subshift of ﬁnite type associated to the

matrix Qis a left shift map T: ΣQ→ΣQi.e., T(xn)n∈Z= (xn+1)n∈Z, where ΣQis the set

of sequences

ΣQ:= {x= (xi)i∈Z:xi∈ {1, ..., k}and Qxi,xi+1 = 1 for all i∈Z};

denote it by (ΣQ, T ). In the case where all entries of the matrix Qare equal to 1, we say

that is the full shift. For simplicity, we denote that ΣQ= Σ.

We call that i0...ik−1is an admissible word if Qin,in+1 = 1 for all 0≤n≤k−2. Let

Ldenote the collection of admissible words, and Lndenote the set of admissible words of

length n. An admissible word of length nis deﬁned as a word x0, x1, ..., xn−1such that

xi∈ {1,2, ..., k}and Qxi,xi+1 = 1.

In the case of a full shift (Σ, T ),Lrepresents the set of all words, while Lnrepresents

the set of words of length n. The length of a word Iin Lis denoted by |I|.

We can deﬁne the n-th level cylinder [I]as follows:

[I] = [i0...in−1] := {x∈Σ : xj=ij∀0≤j≤n−1},

for any i0...in−1∈ Ln.

A cylinder containing x= (xi)i∈Z∈Σof length n∈Nis deﬁned by

[x]n:= (yi)i∈Z∈Σ : xi=yifor all 0≤i≤n−1.

We say that the matrix Qis primitive when there exists n > 0such that all the entries

of Qnare positive. The primitivity of Qis equivalent to the mixing property of the cor-

responding subshift of ﬁnite type (Σ, T ), and such constant nis called the mixing rate of

Σ.

We consider the space Σis endowed with the metric dwhich is deﬁned as follows: For

x= (xi)i∈Z, y = (yi)i∈Z∈Σ, we have

(2.1) d(x, y) = 2−k,

where kis the largest integer such that xi=yifor all |i|< k.

In the two-sided dynamics, we deﬁne the local stable set

loc(x) = {(yn)n∈Z:xn=ynfor all n≥0}

and the local unstable set

loc(x) = {(yn)n∈Z:xn=ynfor all n≤0}.

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arXiv:2210.11574v3[math.DS]22Jul2024ENTROPYSPECTRUMOFLYAPUNOVEXPONENTSFORTYPICALCOCYCLESREZAMOHAMMADPOURID(UPPSALAUNIVERSITY)reza.mohammadpour@math.uu.seAbstract.Inthispaper,westudythesizeofthelevelsetsofallLyapunovexponents.Fortypicalcocycles,weestablishavariationalrelationbetweenthetopologicalentr...

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Entropy spectrum of Lyapunov exponents for typical cocycles

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