CONTINUITY OF THE LYAPUNOV EXPONENTS OF NON-INVERTIBLE RANDOM COCYCLES WITH CONSTANT RANK

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CONTINUITY OF THE LYAPUNOV EXPONENTS OF

NON-INVERTIBLE RANDOM COCYCLES WITH CONSTANT

RANK

PEDRO DUARTE AND CATALINA FREIJO

Abstract. In this paper we establish uniform large deviations estimates of

exponential type and H¨older continuity of the Lyapunov exponents for random

non-invertible cocycles with constant rank.

Contents

1. Introduction 1

2. Statement of results 5

3. An example 10

4. Reduction to invertible cocycles 12

5. Proof of the main results 15

References 17

1. Introduction

In Ergodic Theory the Lyapunov exponent of an i.i.d. matrix valued random

process A1, A2, . . . measures the growth rate of norms of products of matrices in

the process

L= lim

n→∞

nlogkAn· · · A1k.(1.1)

The almost sure existence of this limit was established by H. Furstenberg and

H. Kesten [14, 1960] for ergodic processes, i.e., processes generated by a matrix

valued function over the dynamics of an ergodic measure preserving transforma-

tion. The i.i.d. processes above are generated by locally constant matrix valued

functions on spaces of sequences and Bernoulli shifts on those spaces.

This result marked the beginning of the Theory of Lyapunov exponents of

linear cocycles, which was later greatly ampliﬁed by the Multiplicative ergodic

theorem [18, 1968] of V. I. Oseledets. This theorem provides a global descrip-

tion of all (directional) Lyapunov exponents1of a linear cocycle. An important

1The Lyapunov exponent in (1.1) is sometimes called the top or ﬁrst Lyapunov exponent

because it is the maximum of all directional Lyapunov exponents.

arXiv:2210.14851v1 [math.DS] 26 Oct 2022

2 P. DUARTE AND C. FREIJO

goal of this theory is understanding the stability, or continuity, of the Lyapunov

exponents in terms of the data deﬁning a linear cocycle.

Next we outline some of the main achievements of this theory in the last 60

years regarding the previous goal, for the special class of (invertible) random

linear cocycles, i.e., generated by a locally constant matrix valued function A:

X→GL(d, R) over a Bernoulli or Markov shift σ:X→X. As usual, GL(d, R)

denotes the general linear group of d×dinvertible matrices with real entries.

A ﬁrst milestone of this theory was Furstenberg’s formula in [13, 1963] which

expresses the top Lyapunov exponent via an integral formula involving a so-called

stationary measure of the random linear cocycle.2In [15, 1983] H. Furstenberg

and Y. Kifer used this formula to prove the continuity of the (top) Lyapunov

exponent under a generic irreducibility assumption on the random linear cocycle.

A few years later E. Le Page [16, 1989] proved that the top Lyapunov exponent

is actually H¨older continuous if together with the irreducibility assumption we

assume the simplicity of the ﬁrst Lyapunov exponent. The proof is based on a

spectral method, which exploits the existence of a gap in the spectrum of a certain

Markov operator3that traces the action of the linear cocycle on the projective

space. The existence of a spectral gap is forced by the generic hypothesis of Le

Page’s continuity theorem. This spectral method had been used earlier by Le

Page [19, 1982] to establish limit theorems such as a large deviation principle

and a central limit theorem for random linear cocycles over Bernoulli shifts.

Similar limit theorems for random cocycles over Markov shifts were obtained

by P. Bougerol [7, 1988].

Regarding the problem of general continuity of the Lyapunov exponents (with-

out generic hypothesis), C. Bocker-Neto and M. Viana [6, 2016] proved that this

continuity always holds for random GL(2,R)-valued linear cocycles. A similar re-

sult was announced a few years ago by A. Avila, A. Eskin and M. Viana to hold

for random GL(d, R)-valued cocycles (any d≥2). See Section 10.7 in M. Viana’s

book [28, 2014]. In the same direction, E. Malheiro and M. Viana [17, 2015]

proved the continuity of the Laypunov exponents for random GL(2,R)-valued

linear cocycles over mixing Markov shifts. Still in the framework of GL(2,R)-

valued linear cocycles, these results where extended beyond the class of random

cocycles by L. Bakes, A. Brown and C.Butler [1, 2018] who proved a conjecture by

M. Viana on the general continuity of the Lyapunov exponents for ﬁber-bunched

cocycles over hyperbolic base dynamics.

2The deﬁnition and existence of stationary measures for non-invertible linear cocycles, i.e.,

determined by locally constant functions A:X→Mat(d, R), is problematic. This explains

why a theory of (non-invertible) random linear cocycles has been undeveloped.

3The mentioned spectral gap refers the spectrum of the Markov operator being contained in

some closed disk of radius <1, with the exception of the simple eigenvalue 1 associated with

the constant functions (the operator’s ﬁxed points).

NON-INVERTIBLE RANDOM COCYCLES WITH CONSTANT RANK 3

Concerning regularity, the best we can hope for is that the Lyapunov expo-

nent is an analytic function of a parameter, given a family of linear cocycles

that depends analytically on that parameter. Analyticity of the top Lyapunov

exponent is known to hold in two special cases: First by a theorem of D. Ru-

elle [22, 1982], for uniformly hyperbolic SL(2,R)-cocycles and more generally for

GL(d, R)-cocycles with dominated splitting and a one-dimensional strongly un-

stable direction. Second by a theorem Y. Peres [21, 1991], for random cocycles

generated by locally constant GL(d, R)-valued functions with ﬁnitely many val-

ues, where is shown that any Lyapunov exponent, if simple, is locally analytic as

a function of the probability weights.

An example of a random SL(2,R)-cocycle by B. Haperin, see [23, Appendix

A], shows that the H¨older modulus of continuity of Le Page’s [20, Theorem 1]

is optimal. The recent preprints [4, 3, 2022] deepen the fact illustrated by this

example, showing that if the ratio between the metric entropy of the shift and

the Lyapunov exponent is less than 1 there exists a dichotomy on the regularity

of the Lyapunov exponent, which can either be analytic when the cocycle is

uniformly hyperbolic, or else H¨older continuous with a H¨older exponent which

can not exceed the said ratio when the cocycle is not uniformly hyperbolic.

It is natural to ask about the modulus of continuity of the Lyapunov exponent

as a function of a random cocycle which does not satisfy the assumptions of Le

Page [20, Theorem 1]. Recently E. H. Tall and M. Viana [26, 2020] proved that

for any random GL(2,R)-cocycle, the Lyapunov exponents are always at least

pointwisely log-H¨older continuous4and moreover pointwisely H¨older continuous5

when the Lyapunov exponents are simple. In basically the same setting, random

GL(2,R)-cocycles with ﬁnitely many values, P. Duarte and S. Klein [10, 2020]

proved that the Lyapunov exponents, if simple, are locally weak-H¨older continu-

ous6. In [11, 2019] P. Duarte, S. Klein and M. Santos provided an example of a

random SL(2,R)-cocycle where the assumptions of [20, Theorem 1] fail to hold

and the regularity of the Lyapunov exponent is neither H¨older nor weak-H¨older.

In fact it can not be better than the very bad log3-H¨older modulus of continuity.

In [9, Theorem 3.1], P. Duarte and S. Klein proved that the existence of lo-

cally uniform large deviation estimates of exponential type for a linear cocycle

(deﬁnition in Subsection 2.3) is suﬃcient for the H¨older continuity of the Lya-

punov exponent. This result is used later in the proof of the main theorem. As

4Given a metric space (X, d), a function f:X→Ris said to be pointwisely log-H¨older

continuous at a∈Xif there exists C < ∞such that |f(x)−f(a)| ≤ Clog 1

d(x,a)−1for all x

in some neighborhood of a.

5A function f:X→Ris said to be pointwisely H¨older continuous at a∈Xif there are

C < ∞and 0 < α < 1 such that |f(x)−f(a)| ≤ C d(x, a)αfor all xin some neighborhood of a.

6A function f:X→Ris said to be locally weak-H¨older continuous at a∈Xif there are

C < ∞,c > 0 and 0 < α < 1 such that |f(x)−f(y)| ≤ C e−c(log 1/d(x,y))αfor all x, y in some

neighborhood of a. When α= 1, weak-H¨older becomes H¨older continuity with c-exponent.

4 P. DUARTE AND C. FREIJO

an application, it was proved in [9, Chapter 5] an analogue of Le Page’s H¨older

continuity theorem for random GL(d, R)-cocycles over mixing Markov shifts.

With the single exception of Furstenberg-Kifer [15, Theorem B], all results

described so far assume that the random cocycles are generated by a distribu-

tion law in GL(d, R) which is compactly supported. Recently in [25, 2020], A.

S´anchez and M. Viana proved that the top Lyapunov exponent is upper semi-

continuous, but not continuous, with respect to the Wasserstein distance in the

class of random and non-compactly supported SL(2,R)-cocycles.

Let us now turn to (ﬁber-wise) non-invertible linear cocycles. Many aspects of

the theory of linear cocycles have been extended to such cocycles, namely in what

concerns the Multiplicative ergodic theorem: Blumenthal-Young [5], P. Thieulle

[27], Backes-Maur´ıcio [2]. In [12, 2015], G. Froyland, C. Gonz´alez-Tukman and

A. Quas have proved the stability of the Lyapunov exponents and Oseledets

subspaces under small perturbations with uniform noise. Notice however that

stability is weaker than continuity, which also allows for non-random perturba-

tions.

There are many obstacles to develop a theory for the continuity of the Lya-

punov exponents of random Mat(d, R)-cocycles. First there is no Furstenberg’s

theory about the relation between the ﬁrst Lyapunov exponent and the station-

ary measures of the random cocycle. Via exterior algebra we can always reduce

the study of all Lyapunov exponents to the ﬁrst one. The problem starts with the

fact that non-invertible matrices only induce a partial action on the projective

space. For instance the action of the zero matrix is nowhere deﬁned. Because of

this, stationary measures may not exist, which explains why such a general theory

is impossible. Discontinuities as in [25] can be carried over to this framework.

For instance, given a random GL(2,R)-cocycle with unbounded support, but co-

norm bounded away from 0, which happens to be a discontinuity point of the

ﬁrst Lyapunov exponent, then the inverse cocycle becomes a random and com-

pactly supported Mat(2,R)-valued cocycle where the second Lyapunov exponent

is discontinuous. Another issue is the existence of eventually vanishing cocycles,

i.e., cocycles whose iterates eventually vanish on some set with positive measure,

which by ergodicity must have ﬁrst Lyapunov exponent equal to −∞. We have a

problem if the set of eventually vanishing cocycles is dense in some open set. The

problem is even greater if such cocycles can be approximated by other random

cocycles whose ﬁrst Lyapunov exponent is bounded away from −∞. In this case,

the top Lyapunov exponent becomes nowhere continuous. Such an example is

provided in Section 3.

In order to develop a positive theory we need to consider classes of random

non-invertible cocycles where eventually vanishing cocycles form a closed nowhere

dense set. One such possibility, the one pursued in this manuscript, is to consider

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CONTINUITYOFTHELYAPUNOVEXPONENTSOFNON-INVERTIBLERANDOMCOCYCLESWITHCONSTANTRANKPEDRODUARTEANDCATALINAFREIJOAbstract.InthispaperweestablishuniformlargedeviationsestimatesofexponentialtypeandHoldercontinuityoftheLyapunovexponentsforrandomnon-invertiblecocycleswithconstantrank.Contents1.Introduction12....

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CONTINUITY OF THE LYAPUNOV EXPONENTS OF NON-INVERTIBLE RANDOM COCYCLES WITH CONSTANT RANK

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