MIXED RANDOM-QUASIPERIODIC COCYCLES

MIXED RANDOM-QUASIPERIODIC COCYCLES

AO CAI, PEDRO DUARTE, AND SILVIUS KLEIN

Abstract. We introduce the concept of mixed random - quasiperi-odic linear cocycles. We characterize the ergodicity of the basedynamics and establish a large deviations type estimate for cer-tain types of observables. For the fiber dynamics we prove theuniform upper semicontinuity of the maximal Lyapunov exponent.This paper is meant to introduce a model to be studied in depthin further projects.

1. Introduction

Consider a compactly supported probability measure ν on the groupSLm(R) of m by m matrices with determinant 1 and let

Πn = gn−1 . . . g1 g0

be the random multiplicative process driven by this measure, wheregnn∈Z is an i.i.d. sequence of SLm(R) valued random variables withcommon law ν.

By Furstenberg-Kesten’s theorem (see [5]), the geometric average1n

log ‖Πn‖ converges ν-a.s. to a constant L1(ν) called the maximalLyapunov exponent of the process.

An important example of such a process comes from the study of theAnderson model, the discrete random Schrodinger operator on `2(Z)used in solid state physics to model one dimensional disordered systems(e.g. semiconductors with impurities). This operator is given by

(Hψ)n := −ψn+1 − ψn−1 + wnψn ∀n ∈ Z ,

where ψ = ψnn∈Z ∈ `2(Z) and wnn∈Z is an i.i.d. sequence of realvalued random variables. The corresponding Schrodinger (or eigen-value) equation Hψ = Eψ is equivalent to(

ψn+1

ψn

)=

(wn − E −1

1 0

) (ψnψn−1

)∀n ∈ Z .

Thus it can be solved by means of transfer matrices Πn = gn−1 . . . g1 g0,

where gn :=

(wn − E −1

1 0

)are i.i.d. random matrices.

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2 A. CAI, P. DUARTE, AND S. KLEIN

The behavior of the corresponding Lyapunov exponent as a functionof the energy E (e.g. its positivity or its continuity) is directly pertinentto the study of the spectral properties of the Schrodinger operator.

At the other end of the range of ergodic comportment lies the dis-crete quasiperiodic Schrodinger operator, which in solid state physics isemployed in the description of two dimensional crystal layers immersedin a magnetic field. This operator is given by

(H(θ)ψ)n := −ψn+1 − ψn−1 + vn(θ)ψn ∀n ∈ Z ,where vn(θ) = v(θ + nα) for some continuous function v on the torusTd = (R/Z)d, rationally independent frequency α ∈ Td and phaseθ ∈ Td. The corresponding Schrodinger equation H(θ)ψ = E ψ givesrise to the deterministic (quasiperiodic) multiplicative process

Πn =

(vn−1(θ)− E −1

1 0

). . .

(v1(θ)− E −1

1 0

) (v0(θ)− E −1

1 0

).

Both of these types of multiplicative processes can be studied in themore general framework of linear cocycles. A linear cocycle over anergodic system (X, f, ρ) (referred to as the base dynamics) is a skewproduct map of the form

X × Rm 3 (x, u) 7→ F (x, u) = (f(x), A(x)u) ∈ X × Rm,

where A : X → SLm(R) is a measurable function (referred to as thefiber map). The iterates of F are F n(x, u) = (fn(x), An(x)u), whereAn is the multiplicative process

An(x) = A(fn−1x) . . . A(f(x))A(x) .

Its maximal Lyapunov exponent is defined as before, by Furstenberg-Kesten’s theorem, as the ρ-a.e. limit of 1

nlog ‖An(x)‖.

When the base dynamics is a Bernoulli shift on a space of sequencesand the fiber map A depends only on the zeroth coordinate of thesequence, its iterates An encode a random multiplicative system. Whenthe base dynamics is a torus translation, the iterates of the fiber mapdefine a quasiperiodic multiplicative process.

In this paper we introduce the notion of a mixed random-quasiperiodicmultiplicative process. An important example of such system is relatedto the study of the discrete Schrodinger operator with mixed random-quasiperiodic potential

(Hψ)n = −ψn+1 − ψn−1 + (v(θ + nα) + wn) ψn ∀n ∈ Z.The random part of the potential, given by the sequence wnn∈Z

of i.i.d. random variables, may be regarded as a perturbation of the

MIXED RANDOM-QUASIPERIODIC COCYCLES 3

quasiperiodic part v(θ+nα)n∈Z. A natural question is then to under-stand the influence of this random noise on the behavior of the system.For instance, is the Lyapunov exponent of the quasiperiodic system sta-ble under such random perturbations? This question was posed to usby Jiangong You and it motivated this and several subsequent projects.

A quasiperiodic cocycle can be identified with a pair (α,A), whereα ∈ Td is an ergodic frequency (which defines the base dynamics, atorus translation) andA : Td → SLm(R) is a continuous function (whichinduces the fiber action). Let G be the (metric) space of quasiperiodiccocycles. It turns out that G has a natural group structure, and in fact(G, ) is a topological group. Given a compactly supported measure νon G and an i.i.d. sequence ωnn∈Z of random variables with valuesin the group G and with common law ν, we may interpret the randomproduct of quasiperiodic cocycles

Πn = ωn−1 . . . ω1 ω0

as a mixed random-quasiperiodic multiplicative process.Another (not completely equivalent) way of defining such a process

is to regard it as the iterates of a certain kind of linear cocycle over amixed base dynamics. The latter is a skew-product of a Bernoulli shiftwith a random translation.

This paper is the first in a series of works regarding such mixed pro-cesses. Its purpose is to introduce the main concepts and to establishsome (technical) results, to be used later, a common theme thereof be-ing a certain uniform behavior in the quasiperiodic variable θ (whichis natural, given the unique ergodicity of the torus translation).

The paper is organized as follows. In Section 2 we define the mixedrandom-quasiperiodic base dynamics, characterize its ergodicity (The-orem 2.3) and establish a large deviations type estimate for certainobservables (Theorem 2.4). In Section 3 we formally introduce theconcept of mixed random-quasiperiodic cocycle driven by a measureon the group of quasiperiodic cocycles and establish a uniform upperlarge deviations type estimate (Theorem 3.1). As a consequence, weprove that the maximal Lyapunov exponent is upper semicontinuousas a function of the measure, relative to the Wasserstein distance.

In Section 4 we outline some of the upcoming works on the modelsintroduced here, leading up to the stability under random noise of theLyapunov exponent of a quasiperiodic cocycle. The second and thirdauthors are grateful to Jiangong You for posing this question, thatproved very fruitful, and to Nanjing University for their hospitalityduring an event in 2018 where the conversation took place.


2. The base dynamics

Let (Ω,B) be a standard Borel space. That is, Ω is a Polish space (aseparable, completely metrizable topological space) and B is its Borelσ-algebra.

Let ν ∈ Probc(Ω) be a compactly supported Borel probability mea-sure on Ω. Regarding (Ω, ν) as a space of symbols, we consider thecorresponding (invertible) Bernoulli system

(X, σ, νZ

), where X := ΩZ

and σ : X → X is the (invertible) Bernoulli shift: for ω = ωnn∈Z ∈ X,σω := ωn+1n∈Z. Consider also its non invertible factor on X+ := ΩN.

Let Td = (R/Z)d be the torus of dimension d, and denote by m theHaar measure on its Borel σ-algebra.

Given a continuous function a : Ω→ Td, the skew-product map

f : X × Td → X × Td , f(ω, θ) := (σω, θ + a(ω0)) (2.1)

will be referred to as a mixed random-quasiperiodic (base) dynamics.This map preserves the measure νZ×m and it is the natural extension

of the non invertible map on X+ × Td which preserves the measureνN ×m and is defined by the same expression.

We will study the ergodicity of the mixed random-quasiperiodic sys-tem

(X × Td,f , νZ ×m

). For simplicity, when this holds, we some-

times call the measure ν ergodic, or ergodic with respect to f .We first consider a factor of this system, induced by the function

a : Ω → Td. Regard Σ := Td as a space of symbols equipped with thepush-forward measure µ := a∗ν and consider the skew-product map

f : ΣZ × Td → ΣZ × Td , f(β, θ) := (σβ, θ + β0) , (2.2)

where here σ stands for the Bernoulli shift on the space ΣZ of sequencesβ = βnn∈Z. The function

π : ΩZ × Td → ΣZ × Td, π (ωnn, θ) = (a(ωn)n, θ)

semi-conjugates(ΩZ × Td,f , νZ ×m

)to(ΣZ × Td, f, µZ ×m

). Thus

the second system is a factor of the first, showing in particular thatthe ergodicity of ν implies that of µ. While in general the reverseimplication is not true, in our case it does hold. That is because theaction in the first coordinate is a Bernoulli shift which is mixing.

Proposition 2.1. The measure preserving dynamical system(f , νZ ×m

)is ergodic if and only if

(f, µZ ×m

)is ergodic.

Proof. It is enough to prove the reverse statement.


Recall that a measure preserving dynamical system (X , f, λ) is er-godic if and only if for any ϕ, ψ ∈ L2(X ),

limn→∞

1

n

n−1∑j=0

∫(ϕ f j)ψ dλ =

∫ϕdλ

∫ψ dλ. (2.3)

This equivalent definition of ergodicity will allow us to make use ofthe mixing property of the Bernoulli shift.

Since (2.3) is linear in ϕ and ψ, in order to prove the ergodicity of(X , f, λ) it is enough to find a subset V ⊂ L2(X ) such that LS(V ), thelinear span of V , is dense in L2(X ) and (2.3) holds for any ϕ, ψ ∈ V .

We construct such a set V ⊂ L2(ΩZ × Td) as an increasing limit ofsets Vn of functions depending on a finite number of variables.

Given ϕ : ΩZ → R and ψ : Td → R, denote by ϕ⊗ ψ the function onΩZ × Td defined by ϕ⊗ ψ(ω, θ) := ϕ(ω)ψ(θ).

Then for all n ∈ N, let

Vn :=ϕ⊗ ψ : ϕ ∈ C0

n(ΩZ), ψ ∈ C0(Td),

where C0n(ΩZ) consists of all observables on ΩZ which depend only on

the coordinates (ω−n, · · · , ω0, · · · , ωn). These observables are simplyconditional expectations of absolutely continuous functions with re-spect to the sub-algebra generated by the centered cylinder of length2n + 1. The sequence of sets Vnn≥1 is clearly increasing, so letV :=

⋃∞n=0 Vn.

Recall that the measure µ ∈ Prob(Σ) is the push-forward of ν ∈Prob(Ω) via the map a : Ω→ Σ. We may then consider the disintegra-tion of ν into νββ∈Σ ⊂ Probc(Ω) such that ν =

∫Σνβdµ(β). A direct

computation yields that

νZ =

∫ΣZ

(∏i∈Z

νβi) dµZ(βii∈Z).

We define Av: C0n(ΩZ)→ C0

n(ΣZ) by

(Avϕ)(βii∈Z) :=

∫ΩZϕ(ωii∈Z) d(

∏i∈Z

νβi)(ωii∈Z).

Note that ∫AvϕdµZ =

∫ϕdνZ.

It is straightforward to check that for N > 2n+1 and ϕ, φ ∈ C0n(ΩZ)

we have Av[(ϕ σN)φ] = Av(ϕ σN)Av φ and Av(ϕ σ) = (Avϕ) σ(we use the same symbol σ to denote both the shift on ΩZ and on ΣZ).


Take ϕ1 ∈ C0n(ΩZ), ϕ2 ∈ C0(Td) and ϕ = ϕ1 ⊗ ϕ2. Similarly, take

ψ1 ∈ C0n(ΩZ), ψ2 ∈ C0(Td) and ψ = ψ1 ⊗ ψ2.

For N > 2n+ 1, we have∫ ∫(ϕ fN)ψ dνZ ×m

=

∫ ∫ϕ1(σNω)ϕ2(θ + a(ω0) + · · ·+ a(ωn−1))ψ1(ω)ψ2(θ)dνZ(ω)dm(θ)

=

∫ ∫ϕ1(σNω)ψ1(ω)ϕ2(θ + a(ω0) + · · ·+ a(ωn−1))ψ2(θ)dνZ(ω)dm(θ)

=

∫ ∫Av[(ϕ1 σN)ψ1]ϕ2(θ + β0 + · · ·+ βn−1)ψ2(θ)dµZ(βii∈Z)dm(θ)

=

∫ ∫[(Avϕ1) σN ](Avψ1)ϕ2(θ + β0 + · · ·+ βn−1)ψ2(θ)dµZdm(θ)

=

∫ ∫[(Avϕ1)⊗ ϕ2] fN · [(Avψ1)⊗ ψ2] dµZ ×m,

which converges in the Cesaro sense to∫(Avϕ1)⊗ ϕ2 dµ

Z ×m ·∫

(Avψ1)⊗ ψ2 dµZ ×m (2.4)

since(f, µZ ×m

)is ergodic.

Moreover, we have (same computations for ψ)∫(Avϕ1)⊗ ϕ2 dµ

Z ×m

=

∫Avϕ1 dµ

Z ·∫ϕ2 dm

=

∫ϕ1 dν

Z ·∫ϕ2 dm

=

∫ϕ1 ⊗ ϕ2 dν

Z ×m

=

∫ϕdνZ ×m.

Therefore (2.4) is equal to

∫ϕdνZ×m ·

∫ψ dνZ×m and we conclude

that ∫(ϕ fN)ψ dνZ ×m Cesaro−−−−→

N →∞

∫ϕdνZ ×m ·

∫ψ dνZ ×m

holds for any ϕ, ψ ∈ Vn and n ∈ N thus for any ϕ, ψ ∈ V .This proves that

(f , νZ ×m

)is also ergodic.


Let us consider two basic examples of mixed random-quasiperiodictransformations as in (2.1).

Example 2.1. Given a standard Borel probability measure space (Ω, ν)and a frequency α ∈ Td, let a : Ω → Td be the constant functiona(ω0) ≡ α. Then the corresponding skew-product map f on ΩZ × Tdis given by

f(ω, θ) = (σω, θ + α) .

Thus the system (f , νZ×m) is just the product between the Bernoullishift σ and the torus translation by α, which we denote by τα.

Moreover, since µ = a∗ν = δα (the Dirac measure centered at α), itsfactor (f, µZ ×m) as defined above is clearly isomorphic to the torustranslation (τα,m). By Proposition 2.1, (f , νZ ×m) is ergodic if andonly if (τα,m) is ergodic, which is of course well known.

Example 2.2. Given a standard Borel probability measure space (S, ρ)and µ ∈ Prob(Td), let Ω := Td × S, ν := µ × ρ and let a : Ω → Td bethe projection in the first coordinate, a(β, b) = β. It clearly holds thata∗ν = a∗(µ× ρ) = µ.

The corresponding skew-product map on ΩZ × Td is given by

f(ωn, θ) = (ωn+1, θ + a(ω0)) ,

while its factor on ΣZ × Td is

f(βn, θ) = (βn+1, θ + β0) .

By Proposition 2.1, ν is ergodic with respect to f if and only if µ isergodic with respect to f .

2.1. Stochastic dynamical systems. We introduce some general con-cepts that will be used throughout the paper.

Given a metric space (M,d), denote by C0(M), Cb(M), Lip(M), re-spectively, the spaces of continuous functions, continuous and boundedfunctions and Lipschitz continuous real valued functions on M . Let‖g‖0 denote the uniform norm of a function g ∈ Cb(M) and let ‖g‖Lip

denote the best Lipschitz constant of a function g ∈ Lip(M).The following Urysohn type lemma will be needed in the sequel.

Lemma 2.2. Let (M,d) be a metric space and let ν be a Borel proba-bility measure in M . Given a closed set L ⊂M and ε > 0 there are anopen set D ⊃ L such that ν(D) < ν(L) + ε and a Lipschitz continuousfunction g : M → [0, 1] such that 1L ≤ g ≤ 1D.

Proof. For every δ > 0 let Lδ := x ∈M : d(x, L) < δ be the openδ-neighborhood of L. Since L is closed we have that

⋂δ>0 Lδ = L.


Then ν(Lδ)→ ν(L) as δ → 0, so there is δ0 = δ0(L, ε, ν) > 0 such thatν(Lδ0) < ν(L) + ε.

Let D := Lδ0 and note that d(L,D

)= d

(L,Lδ0

)≥ δ0 > 0. One

can then easily verify that the function g : M → R,

g(x) :=d(x,D)

d(x,D) + d(x, L)

is Lipschitz continuous with ‖g‖Lip ≤1δ0

, while clearly 1L ≤ g ≤ 1D.The main point here is that the closed set L need not be compact, as

its distance to the closed set D is already bounded away from zero.

Let Prob(M) denote the space of Borel probability measures on Mand define the weak* convergence of a sequence νn → ν by

∫φdνn →∫

φdν for all φ ∈ Cb(M).Furthermore, let

Prob1(M) := ν ∈ Prob(M) :

∫M

d(x, x0) dν(x) <∞ ,

where x0 ∈ M is an arbitrary point (whose choice is of course incon-sequential). Note that Probc(M), the space of compactly supportedBorel probability measures on M , is contained in Prob1(M).

If M is a compact metric space then the weak* convergence definesthe weak topology on Probc(M) = Prob1(M) = Prob(M) and thistopology is compact and metrizable. If M is a (more general) Polishmetric space (thus not necessarily compact), the weak topology onProb1(M) is defined by the weak* convergence νn → ν together withthe convergence

∫d(x, x0) dνn →

∫d(x, x0) dν for some (and hence all)

x0 ∈M . Then Prob1(M) is itself a Polish space.In either case, consider the Wasserstein (or Kantorovich-Rubinstein)

distance W1 in the space Prob1(M), where

W1(ν, ν ′) := sup

∫g d(ν − ν ′) : g ∈ Lip(M), ‖g‖Lip ≤ 1

.

It is well known that this distance metrizes the weak topology onProb1(M), see [11, Chapter I.6] for this and all other related conceptsmentioned above.

A stochastic dynamical system (SDS) on M (also called a randomwalk in [6]) is any continuous map K : M → Prob(M), x 7→ Kx. AnSDS K on M induces a bounded linear operator (called the Markovoperator) QK : Cb(M)→ Cb(M) defined by

(QKϕ)(x) :=

∫M

ϕ(y) dKx(y).


It also induces the adjoint operator Q∗K : Prob(M)→ Prob(M) of QKcharacterized by

Q∗Kν = K ∗ ν :=

∫M

Kx dν(x).

A measure ν ∈ Prob(M) is called K-stationary if Q∗Kν = ν. We denoteby ProbK(M) the convex and compact subspace of all K-stationaryprobability measures on M .

Let (G, ·) be a topological group acting on M from the left. Denoteby τg : M →M the action on M by g ∈ G, that is, τg(x) = gx.

Given µ ∈ Probc(G) and ν ∈ Probc(M), the convolution µ ∗ ν ∈Probc(M) is given by

µ ∗ ν(E) :=

∫M

∫G

1E(gx) dµ(g)dν(x)

for any Borel set E ⊂M .Then

µ ∗ ν =

∫G

(τg)∗ ν dµ(g) ,

where (τg)∗ ν is the push-forward probability measure

(τg)∗ ν(E) := ν(τ−1g E

)= ν

(g−1E

).

A probability measure µ ∈ Probc(G) determines an SDS on M by

M 3 x 7→ µ ∗ δx =

∫G

δgx dµ(g) ∈ Probc(M) .

The associated Markov operator Qµ : Cb(M)→ Cb(M) is given by

(Qµφ) (x) =

∫M

φ(y) dµ ∗ δx(y) =

∫G

φ(gx) dµ(g) .

Moreover, its dual operator Q∗µ : Probc(M)→ Probc(M) is

Q∗µν =

∫M

µ ∗ δx dν(x) = µ ∗ ν .

Let

Probµ(M) := ν ∈ Probc(M) : µ ∗ ν = νbe the set of µ-stationary measures on M , that is, the fixed points ofthe dual Markov operator Q∗µ.

Given such a µ-stationary measure ν, any observable φ : M → R forwhich

(Qµφ) (x) = φ(x) for ν a.e. x ∈Mis called a ν-stationary observable.


Specializing to G = M = Td seen as an additive group, for α, θ ∈ Tdand µ ∈ Prob(Td) we have τα(θ) = θ + α,

(Qµφ) (θ) =

∫Tdφ(θ + α) dµ(α)

and

Q∗µν(E) =

∫Td

(τα)∗ ν(E) dµ(α) =

∫Tdν(τ−1α E

)dµ(α)

for any Borel measurable set E ⊂ Td.Let m be the Haar measure on Td. Note that m is µ-stationary (since

it is translation invariant).Finally, given any k ∈ Zd, we define the corresponding Fourier coef-

ficient of the measure µ by

µ(k) :=

∫Σ

e2πi〈k,α〉 dµ(α).

2.2. Ergodicity of the base dynamics. Proposition 2.1 reduces thestudy of the ergodicity of a skew product map like (2.1) to that of itsfactor (2.2). We will then study the latter.

The following result provides various characterizations of the ergod-icity of the base transformation. Some of them, e.g. (4) and (5) canalso be deduced from Anzai’s theorem (see [9, Theorem 4.8]) on theergodicity of general skew products.

Theorem 2.3. Let µ ∈ Prob(Σ) where Σ = Td, and consider the skewproduct map on ΣZ × Td given by f(βi, θ) = (σβi, θ + β0). Thefollowing statements are equivalent:

(1) f is ergodic w.r.t. µZ ×m;(2) f is ergodic w.r.t. µN ×m;(3) Every m-stationary observable ϕ ∈ L∞(Td) is constant m-a.e.;(4) µ(k) 6= 1 for every k ∈ Zd \ 0;(5) For every k ∈ Zd \ 0 there exists α ∈ S such that 〈k, α〉 /∈ Z;(6) Td = ∪n≥1Sn where S = supp (µ) and Sn := S+Sn−1 ∀n ≥ 2;(7) m is the unique µ-stationary measure in Prob(Td),(8) limn→+∞

1n

∑n−1j=0 (Qjµϕ)(θ) =

∫Td ϕdm, ∀ θ ∈ Td ∀ϕ ∈ C0(Td).

Proof. (1) ⇒ (2) holds trivially because f in (2) is a factor f in (1),i.e., because of the commutativity of the following diagram of measurepreserving transformations.

Xf−−−→ X

π

y yπX+ f−−−→ X+


Conversely, (2) ⇒ (1) holds by Lemma 5.3.1 in [7].The equivalence (2) ⇔ (3) follows from Proposition 5.13 in [10].

Given a bounded measurable function ϕ : Td → C, we have ϕ ∈L2(Td,m). Consider its Fourier series

ϕ =∑k∈Zd

ϕ(k) ek with ek(θ) := e2πi〈k,θ〉.

A simple calculation shows that

Qµϕ =∑k∈Zd

µ(k) ϕ(k) ek.

(3) ⇒ (4): If µ(k) = 1 for some k ∈ Zd \ 0, then ek is a nonconstant m-stationary observable. In other words, if (4) fails then sodoes (3).

(4) ⇒ (3): Given ϕ m-stationary, comparing the two Fourier devel-opments above, for all k ∈ Zd µ(k) ϕ(k) = ϕ(k) ⇔ ϕ(k) (µ(k)−1) =0. By (4) we then get ϕ(k) = 0 for all k ∈ Zd \ 0, which implies thatϕ = ϕ(0) is m-a.e. constant. This proves (3).

Since µ(k) is an average of a continuous function with values on theunit circle, we have

µ(k) = 1 ⇔ e2πi〈k,α〉 = 1, ∀α ∈ S ⇔ 〈k, α〉 ∈ Z, ∀α ∈ S.

This proves that (4) ⇔ (5).

(5) ⇒ (6): Let H = ∪n≥1Sn and assume that H 6= Td. By definitionH is a subsemigroup of Td. By Poincare recurrence theorem, H is alsoa group. By Pontryagin’s duality for locally compact abelian groups,there exists a non trivial character ek : Td → C which contains H inits kernel. In particular this implies that there exists k ∈ Zd \ 0 suchthat 〈k, β〉 ∈ Z for all β ∈ S. This argument shows that if (6) failsthen so does (5).

(6) ⇒ (5): Assume that (5) does not hold, i.e., for some k ∈ Zd\0we have 〈k, α〉 ∈ Z for all α ∈ S. Then ek is a non trivial character ofTd and H := θ ∈ Td : ek(θ) = 1 is a proper sub-torus, i.e. a compactsubgroup of Td. The assumption implies that S ⊂ H, and since H is agroup, Sn ⊂ H, ∀n ≥ 1. This proves that (6) fails.

Since the adjoint operator Q∗µ : Prob(Td) → Prob(Td) satisfies

Q∗µπ = µ ∗ π, denoting by µ∗j := µ ∗ · · · ∗ µ the j-th convolutionpower of µ, we have (Q∗µ)nδ0 = µ∗n ∀n ∈ N.


Lemma 2.3. Any sublimit of the sequence πn := 1n

∑n−1j=0 µ

∗j is a µ-stationary measure.

Proof. Given ϕ ∈ C0(Td),

〈Qµϕ− ϕ, πn〉 =1

n

n−1∑j=0

〈Qµϕ− ϕ, (Q∗µ)jδ0〉

=1

n

n−1∑j=0

(Qj+1µ ϕ)(0)− (Qjµϕ)(0)

=1

n((Qnµϕ)(0)− ϕ(0)) = O(

1

n).

Hence, if π ∈ Prob(Td) is a sublimit of πn, taking the limit along thecorresponding subsequence of integers we have

〈ϕ,Q∗µπ − π〉 = 〈Qµϕ− ϕ, π〉 = 0,

which implies that Q∗µπ = π.

(2) ⇒ (8): By ergodicity of f w.r.t. µN ×m and Birkhoff ErgodicTheorem, given ϕ ∈ C0(Td) there exists a full measure set of (ω, θ) ∈SN × Td with

limn→+∞

1

n

n−1∑j=0

ϕ(θ + τ j(ω)) =

∫ϕdm,

where τ j(ω) = ω0 + · · · + ωj−1 and ω = ωjj∈N. Hence there exists aBorel set B ⊂ Td with m(B) = 1 such that, applying the DominatedConvergence Theorem, we have for all θ ∈ B,

limn→+∞

1

n

n−1∑j=0

(Qjµϕ)(θ) =

∫ϕdm.

The set B depends on the continuous function ϕ, but since the spaceC0(Td) is separable we can choose this Borel set B so that the previouslimit holds for every θ ∈ B and ϕ ∈ C0(Td). This implies the followingweak* convergence in Prob(Td):

limn→+∞

1

n

n−1∑j=0

(Q∗µ)jδθ = m.


Given any θ′ /∈ B take θ ∈ B. Convolving both sides on the right byδθ′−θ we get

limn→+∞

1

n

n−1∑j=0

(Q∗µ)jδθ′ = limn→+∞

1

n

n−1∑j=0

µ∗j ∗ δθ ∗ δθ′−θ = m ∗ δθ′−θ = m,

which proves (8).(8) ⇒ (7): If there exists η 6= m in Probµ(Td), then there exists at

least one more ergodic measure ζ 6= m such that ζ is an extreme pointof Probµ(Td). Choosing ϕ ∈ C0(Td) such that

∫ϕdζ 6=

∫ϕdm, by

Birkhoff Ergodic Theorem there exists θ ∈ Td such that

limn→+∞

1

n

n−1∑j=0

(Qjµϕ)(θ) =

∫ϕdζ 6=

∫ϕdm.

which contradicts (8).(7) ⇒ (6): Consider the compact subgroup H := ∪n≥1Sn. If (6)

fails then H 6= Td and by Lemma 2.3 we can construct a stationarymeasure π ∈ Probµ(Td) with supp (π) ⊂ H. This shows that π 6= mand hence there is more than one stationary measure.

Proposition 2.4. If f is ergodic w.r.t. µZ ×m then the convergencein item (8) of Proposition 2.3 holds uniformly in θ ∈ Td.

Proof. We prove it by contradiction. Assume there ∃ ε > 0, ∃nk →∞and ∃ θk ∈ Td, k ∈ N+ such that∣∣∣∣∣ 1

nk

nk−1∑j=0

(Qjµϕ)(θk)−∫Tdϕdm

∣∣∣∣∣ > ε.

Since Td is compact, we can assume θk → θ for some θ ∈ Td. Writing

1

nk

nk−1∑j=0

(Qjµϕ)(θ)−∫ϕdm =

1

nk

nk−1∑j=0

(Qjµϕ)(θ)− 1

nk

nk−1∑j=0

(Qjµϕ)(θk) +1

nk

nk−1∑j=0

(Qjµϕ)(θk)−∫ϕdm


we have ∣∣∣∣∣ 1

nk

nk−1∑j=0

(Qjµϕ)(θ)−∫ϕdm

∣∣∣∣∣ ≥∣∣∣∣∣ 1

nk

nk−1∑j=0

(Qjµϕ)(θk)−∫ϕdm

∣∣∣∣∣−∣∣∣∣∣ 1

nk

nk−1∑j=0

(Qjµϕ)(θ)− 1

nk

nk−1∑j=0

(Qjµϕ)(θk)

∣∣∣∣∣ ≥ ε− ε

2≥ ε

2,

where the second inequality is due to the definition of Qjµϕ and to the

uniform continuity of ϕ on Td. This contradicts (8).

2.3. Uniform convergence of the Birkhoff sums. We return tothe study our original base dynamics f : X × Td → X × Td definedby (2.1), where X = ΩZ. Since the ergodicity of f is equivalent to thatof its factor f , and since these two maps share similar expressions, tosimplify notations, from now on we let f refer to either one of them.

Under the ergodicity assumption, we prove that for a full measureset of points ω ∈ X, given any continuous observable φ : X × Td → R,the corresponding Birkhoff time averages converge to the space averageuniformly in θ ∈ Td.

Lemma 2.5. Let ν ∈ Probc(Ω) and assume that f is ergodic w.r.t.νZ × m. There is a full measure set X ′ ⊂ X such that given anyobservable φ ∈ C0(X × Td), for all ω ∈ X ′ we have

limn→+∞

1

n

n−1∑j=0

φ(f j(ω, θ)) =

∫φ d(νZ ×m)

with uniform convergence in θ ∈ Td.

Proof. Let S := supp ν and X := SZ. Since X is compact, σ-invariantand νZ(Z) = 1, in order to prove the above convergence of the Birkhoffmeans for observables φ ∈ C0(X × Td) we may simply consider theirrestrictions to the compact, metrizable space X × Td. As C0(X × Td)is separable, it admits a countable and dense subset φj : j ≥ 1.

Denote by

Bj :=

(ω, θ) ∈ X × Td : limn→+∞

1

n

n−1∑i=0

φj(fi(ω, θ)) =

∫φj d(νZ ×m)

.


If we denote B =⋂j≥1 Bj, then by the Birkhoff Ergodic Theorem

(νZ × m)(B) = 1. Thus for m-a.e. θ ∈ Td, νZ(Bθ) = 1 where Bθ =ω ∈ X : (ω, θ) ∈ B.

Fix θ0 ∈ Td such that νZ(Bθ0) = 1. For νZ-a.e. ω (in fact forω ∈ Bθ0 ⊂ X ), we have that for all φ ∈ C0(X × Td) and ε > 0, thereexists n0 such that for all n ≥ n0 and for j large enough∣∣∣∣∣ 1n

n−1∑i=0

φ(f i(ω, θ0))−∫φ d(νZ ×m)

∣∣∣∣∣≤

∣∣∣∣∣ 1nn−1∑i=0

φ(f i(ω, θ0))− 1

n

n−1∑i=0

φj(fi(ω, θ0))

∣∣∣∣∣+∣∣∣∣∣ 1nn−1∑i=0

φj(fi(ω, θ0))−

∫φj d(νZ ×m)

∣∣∣∣∣+∣∣∣∣∫ φj d(µZ ×m)−∫φ d(νZ ×m)

∣∣∣∣≤ ε

3+ε

3+ε

3≤ ε,

where in the second inequality we used the density of φj : j ≥ 1 inC0(X ×Td) and the definition of B. This shows that for νZ-a.e. ω, thefollowing weak* convergence holds:

1

n

n−1∑i=0

δf i(ω,θ0) → νZ ×m. (2.5)

This in fact holds for any θ ∈ Td (not just for the given θ0) by the m-invariance of the torus translation. Indeed, the action of Td on X ×Tdgiven by θ · (ω, θ′) = (ω, θ + θ′) induces a convolution of measures anda direct computation shows that δθ−θ0 ∗ δfj(ω,θ0) = δfj(ω,θ) for all j ≥ 1and δθ−θ0 ∗ (νZ×m) = νZ×m. Then by (2.5) and the weak* continuityof the convolution operation, for νZ-a.e. ω and every θ ∈ Td,

1

n

n−1∑i=0

δf i(ω,θ) → νZ ×m.

This is equivalent to saying that for νZ-a.e. ω (that is, for ω ∈ Bθ0),for all φ ∈ C0(X × Td) and all θ ∈ Td,

1

n

n−1∑j=0

φ(f j(ω, θ))→∫φ d(νZ ×m).


We prove the uniform convergence in θ by contradiction. Assumethat there are ω ∈ Bθ0 , ε > 0, nk →∞ and θk ∈ Td for all k ≥ 1 suchthat ∣∣∣∣∣ 1

nk

nk−1∑j=0

φ(f j(ω, θk))−∫φ d(µZ ×m)

∣∣∣∣∣ ≥ ε.

Since Td is compact, by passing to a subsequence we may assumethat θk → θ. Then for k sufficiently large we have:∣∣∣∣∣ 1

nk

nk−1∑j=0

φ(f j(ω, θ))−∫φ d(νZ ×m))

∣∣∣∣∣≥

∣∣∣∣∣ 1

nk

nk−1∑j=0

φ(f j(ω, θk))−∫φ d(νZ ×m))

∣∣∣∣∣−

∣∣∣∣∣ 1

nk

nk−1∑j=0

φ(f j(ω, θ))− 1

nk

nk−1∑j=0

φ(f j(ω, θk))

∣∣∣∣∣≥ε− ε

2≥ ε

2.

The second inequality follows from the fact that, for k large enough,

|φ(ω′, θk)− φ(ω′, θ)| < ε

2∀ω′ ∈ X ,

which is due to the uniform continuity of φ on the compact set X ×Td.This contradicts the pointwise convergence for θ.

The next result establishes a large deviations type estimate over er-godic mixed random-quasiperiodic systems, for continuous observablesthat depend on finitely many coordinates. The estimate is uniform inthe quasiperiodic variable θ and also in the measure determining therandom variable.

Theorem 2.4. Let ν0 ∈ Probc(Ω) be an ergodic measure w.r.t. f andlet φ ∈ Cb(X×Td) be an observable that depends on a finite number ofcoordinates of ω ∈ X. Given any ε > 0, there are δ = δ(ε, ν0, φ) > 0,n = n(ε, ν0, φ) ∈ N and c = c(ε, ν0, φ) > 0 such that for all ν ∈Probc(Ω) with W1(ν, ν0) < δ, for all θ ∈ Td and for all n ≥ n we have

νZ

ω ∈ X :

∣∣∣∣∣ 1nn−1∑j=0

φ(f j(ω, θ))−∫X×Td

φ d(νZ ×m)

∣∣∣∣∣ ≥ ε

< e−cn .

(2.6)A similar estimate also holds with the integral in (2.6) taken with

respect to the fixed measure νZ0 ×m.


Proof. We use a stopping time argument where the times are chosenuniformly in θ; this way we decouple the variables ω and θ and reducethe problem to a concentration inequality over the Bernoulli shift σ foran observable that depends on finitely many random coordinates.

Fix ε > 0. It is easy to see that if we established (2.6) with someconstant a(φ) instead of

∫φ d(νZ ×m), then we would have∣∣∣∣a(φ)−

∫X×Td

φ d(νZ ×m)

∣∣∣∣ < ε+ ‖φ‖0 e−cn < 2ε

for n large enough, which would therefore imply (2.6) as written. Wewill then establish the estimate with a(φ) :=

∫φ d(νZ0 ×m).

Note also that replacing φ by −φ, it suffices to prove the upperbound in (2.6), namely that for ω outside an exponentially small setwith respect to the νZ measure, and for all θ ∈ Td we have

1

n

n−1∑j=0

φ(f j(ω, θ)) < a(φ) + ε . (2.7)

Finally, replacing φ by φ− inf φ, we may assume that φ ≥ 0.Using Lemma 2.5, for νZ0 -a.e. ω ∈ X we can define n(ω) = n(ω, ε)

to be the first integer such that for all θ ∈ Td,

1

n(ω)

n(ω)−1∑j=0

φ(f j(ω, θ)) < a(φ) + ε .

Given m ∈ N, let

Um := ω ∈ X : n(ω) ≤ m

=m⋃k=1

ω ∈ X :

1

k

k−1∑j=0

φ f j(ω, θ)) < a(φ) + ε ∀θ ∈ Td.

Since φ and f are continuous and Td is compact, the set Um isopen. Moreover, as the sequence of sets Umm≥1 increases to a full

νZ0 -measure set, there is N = N(ε, ν0, φ) such that νZ0 (UN) < ε.Note that since the observable φ depends on a finite number (say k0)

of coordinates, the set UN is determined by k := k0 + N coordinates,where k = k(ε, ν0, φ). The same of course holds for its complementUN , which is a closed set. Let L ⊂ Ωk be the projection of UN in thek coordinates on which it depends. Then L must be a closed set andfor any ν ∈ Probc(Ω) we have νZ(UN) = νk(L).

We claim that if a measure ν is chosen sufficiently close to ν0 relativeto the Wasserstein distance, we can ensure that the νZ measure of UN


is of order ε as well. Indeed, applying Lemma 2.2 to the closed setL ⊂ Ωk, there are an open set D ⊃ L such that

νk0 (D) ≤ νk0 (L) + ε = νZ0 (UN) + ε < 2ε

and a Lipschitz continuous function g : Ωk → [0, 1] such that 1L ≤ g ≤1D and ‖g‖Lip = C = C(ε, L, k) = C(ε, ν0, φ).

It is easy to see that for any ν ∈ Probc(Ω) we have

W1(νk, νk0 ) ≤ kW1(ν, ν0) ,

so ∣∣∣∣∫Ωkg d(νk − νk0 )

∣∣∣∣ ≤ CW1(νk, νk0 ) ≤ CkW1(ν, ν0).

Then

νZ(UN) = νk(L) =

∫Ωk1L dν

k ≤∫

Ωkg dνk ≤

∫Ωkg dνk0 + CkW1(ν, ν0)

≤∫

Ωk1D dν

k0 + CkW1(ν, ν0) = νk0 (D) + CkW1(ν, ν0) < 3ε ,

provided that W1(ν, ν0) < δ =: εCk

.

By design, for all ω ∈ UN we have 1 ≤ n(ω) ≤ N and for all θ ∈ Td,n(ω)−1∑j=0

φ(f j(ω, θ)) ≤ n(ω) a(φ) + n(ω) ε . (2.8)

Fix any ω = ωjj∈Z ∈ X and define inductively a sequence ofintegers nk = nk(ω)k≥1 as follows.

If ω ∈ UN then n1 := n(ω), otherwise n1 := 1.If σn1ω ∈ UN then n2 := n(σn1ω), otherwise n2 := 1.If, for k ≥ 1, we have σnk+...+n1ω ∈ UN then nk+1 := n(σnk+...+n1ω),

otherwise nk+1 := 1. Note that 1 ≤ nk ≤ N for all k ≥ 1.Using (2.8) (and the fact that φ ≥ 0), for all θ ∈ Td, the Birkhoff

sum of length n1 with starting phase (ω, θ) has the bound

n1−1∑j=0

φ(f j(ω, θ)) ≤ n1a(φ) + n1 ε+ ‖φ‖0 1UN

(ω) .

Similarly, the Birkhoff sum of length n2 with starting phase fn1(ω, θ) =(σn1ω, θ + a(ω0) + . . .+ a(ωn1−1)) has the bound

n2−1∑j=0

φ(f j+n1(ω, θ)) ≤ n2a(φ) + n2 ε+ ‖φ‖0 1UN

(σn1ω) .


In general, for k ≥ 1, the Birkhoff sum of length nk+1 with startingphase fnk+...+n1(ω, θ) = (σnk+...+n1ω, θ + a(ω0) + . . . + a(ωnk+...+n1−1))has the bound

nk+1−1∑j=0

φ(f j+nk+...+n1(ω, θ)) ≤ nk+1a(φ) +nk+1ε+ ‖φ‖0 1UN

(T nk+...+n1ω).

Let n = n(ε, µ, φ) := N max‖φ‖0ε, 1

, so n ≥ N ≥ n1. Fix any

n ≥ n. Since n1 < n1 + n2 < . . . < n1 + . . . + nk < . . . , there is p ≥ 1such that n = n1 + . . . np +m, where 0 ≤ m < np+1 ≤ N .

It follows that

n−1∑j=0

φ(f j(ω, θ)) =

n1+...+np−1∑j=0

φ(f j(ω, θ)) +m−1∑j=0

φ(f j+n1+...+np(ω, θ))

=

p−1∑k=0

nk+1−1∑j=0

φ(f j+nk+...+n1(ω, θ))

+m−1∑j=0

φ(f j+n1+...+np(ω, θ)) ,

hence

n−1∑j=0

φ(f j(ω, θ)) ≤ (n1 + . . .+ np)a(φ) + (n1 + . . .+ np)ε

+ ‖φ‖0

p−1∑k=0

1UN

(σnk+...+n1ω) +m ‖φ‖0

≤ na(φ) + nε+n−1∑j=0

1UN

(σjω) +N ‖φ‖0

< na(φ) + 2nε+n−1∑j=0

1UN

(σjω) .

We obtained the following: for all ω ∈ X, θ ∈ Td and n ≥ n,

1

n

n−1∑j=0

φ(f j(ω, θ)) < a(φ) + 2ε+1

n

n−1∑j=0

1UN

(σjω) . (2.9)

It remains to estimate the Birkhoff average over the Bernoulli shift ofthe function 1U

n. Since 1U

ndepends on k coordinates, its n-th Birkhoff


average depends on n+k−1 coordinates. Hence the following functionis well defined (and it is measurable):

h : Ωn+k−1 → R, h(x0, . . . , xn+k−2) :=1

n

n−1∑j=0

1UN

(σjω) ,

where ω = ωj∈Z ∈ X with ω0 = x0, . . . , ωn−k−2 = xn−k−2.Because of the dependence of 1U

Non k coordinates, the function h

satisfies the following bounded differences property:

|h(x0, . . . , xi−1, xi, xi+1, . . . , xn+k−2)

−h(x0, . . . , xi−1, x′i, xi+1, . . . , xn+k−2)| ≤ 2k

n‖h‖∞ =

2k

n.

Then by McDiarmid’s inequality (see [8, Theorem 3.1]), for any prob-ability measure ν on Ω there is an exceptional set Bn ⊂ Ωn+k−1 with

νn+k−1(Bn) < e−ε2

2k2n, so that for (ω0, . . . , ωn+k−2) /∈ Bn we have:

h(ω0, . . . , ωn+k−2)−∫h dνn+k−1 < ε .

Clearly ∫h(ω0, . . . , ωn+k−2) dνn+k−1(ω0, . . . , ωn+k−2)

=

∫1

n

n−1∑j=0

1UN

(σjω) dνZ(ω)

=

∫1U

N(ω) dνZ(ω) = νZ(UN) < 3ε ,

which when combined with (2.9) implies (2.7).

3. The fiber dynamics

In this section we formally introduce the concept of mixed random-quasiperiodic cocycle, present a motivating example and study the up-per semicontinuity of its maximal Lyapunov exponent.

3.1. The group of quasiperiodic cocycles. A quasiperiodic cocycleis a skew-product map of the form

Td × Rm 3 (θ, v) 7→ (τα(θ), A(θ)v) ∈ Td × Rm ,

where τα(θ) = θ+α is a translation on Td by a rationally independentfrequency α ∈ Td and A ∈ C0(Td, SLm(R)) is a continuous matrixvalued function on the torus.


This cocycle can thus be identified with the pair (α,A). Considerthe set

G = G(d,m) := Td × C0(Td, SLm(R))

of all quasiperiodic cocycles.This set is a Polish metric space when equipped with the product

metric (in the second component we consider the uniform distance).The space G is also a group, and in fact a topological group, with thenatural composition and inversion operations

(α,A) (β,B) := (α + β, (A τβ)B)

(α,A)−1 := (−α, (A τ−α)−1) .

Given ν ∈ Probc(G) let ω = ωnn∈Z, ωn = (αn, An) be an i.i.d.sequence of random variables in G with law ν. Consider the corre-sponding multiplicative process in the group G

Πn = ωn−1 . . . ω1 ω0

=(αn−1 + . . .+ α1 + α0, (An−1 ταn−2+...+α0) . . . (A1 τα0)A0

).

In order to study this process in the framework of ergodic theory, wemodel it by the iterates of a linear cocycle.

3.2. Mixed random-quasiperiodic cocycles. Given ν ∈ Probc(G),let Ω ⊂ G be a closed subset (thus a a Polish space as well) suchthat Ω ⊃ supp ν. Depending on what will be convenient in a specificsituation, Ω can be the entire space G, or a compact set, say supp ν or,for a given constant L < ∞, the set GL := (α,A) ∈ G : ‖A‖0 ≤ L.In any case, the choice of the set Ω ⊃ supp ν will not influence thedefinitions and results to follow.

We regard (Ω, ν) as a space of symbols and consider, as before, theshift σ on the space X := ΩZ of sequences ω = ωnn∈Z endowed withthe product measure νZ and the product topology (which is metriz-able). The standard projections

a : Ω→ Td, a(α,A) = α

A : Ω→ C0(Td, SLm(R)), A(α,A) = A

determine the linear cocycle F = F(a,A) : X ×Td×Rm → X ×Td×Rm

defined by

F (ω, θ, v) := (σω, θ + a(ω0),A(ω0)(θ) v) .

The non-invertible version of this map, with the same expression, isdefined on X+ × Td × Rm, where X+ = ΩN.


Thus the base dynamics of the cocycle F is the mixed random-quasiperiodic map

X × Td 3 (ω, θ) 7→ (σω, θ + a(ω0)) ∈ X × Td,

while the fiber action is induced by the map

X × Td 3 (ω, θ) 7→ A(ω, θ) =: A(ω0)(θ) ∈ SLm(R).

The skew-product F will then be referred to as a mixed random-quasiperiodic cocycle.

For ω = ωnn∈Z ∈ X and j ∈ N consider the composition of randomtranslations

τ jω := τa(ωj−1) . . . τa(ω0) = τa(ωj−1)+...+a(ω0) = τa(ωj−1...ω0) .

The iterates of the cocycle F are then given by

F n(ω, θ, v) = (σnω, τnω (θ), An(ω)(θ)v) ,

where

An(ω) = A (ωn−1 . . . ω1 ω0)

=(A(ωn−1) τn−2

ω

). . .(A(ω1) τ 0

ω

)A(ω0) .

Thus An(ω) can be interpreted as a random product of quasiperiodiccocycles. For convenience we also denote An(ω, θ) := An(ω)(θ).

By the subadditive ergodic theorem, the limit of1

nlog ‖An(ω)(θ)‖

as n→∞ exists for νZ×m a.e. (ω, θ) ∈ X ×Td. If the base dynamicsf is ergodic w.r.t. νZ ×m, then this limit is a constant that dependsonly on the measure ν and it is called the maximal Lyapunov exponentof the cocycle F , which we denote by L1(ν). Thus

L1(ν) = limn→∞

1

nlog ‖An(ω)(θ)‖ for νZ ×m a.e. (ω, θ)

= limn→∞

∫X×Td

1

nlog ‖An(ω)(θ)‖ d(νZ ×m) .

An important problem, to be studied more in depth in future projectsconcerns the continuity properties of the map ν 7→ L1(ν).

Remark 3.1. An alternative, somewhat more particular way to de-fine mixed random-quasiperiodic cocycles is the following. Fix an ab-stract space of symbols (Ω, ρ) (where Ω is a Polish metric space andρ ∈ Probc(Ω)) and a continuous function a : Ω→ Td. Consider the cor-responding mixed quasiperiodic base dynamics (X ×Td, f, ρZ×m) de-fined in Section 2, where X = ΩZ. A continuous function A : X×Td →


SLm(R) that depends only the coordinate ω0 of ω ∈ X and on θ ∈ Tddetermines the linear cocycle F = F(a,A) over f given by

F (ω, θ, v) = (f(ω, θ), A(ω, θ)v) = (σω, θ + a(ω0),A(ω, θ)v) . (3.1)

Note that since A depends only on the coordinate ω0 of ω ∈ X andon θ ∈ Td, it can be identified with the map

Ω 3 ω0 7→ A(ω0) ∈ C0(Td, SLm(R)), A(ω0)(θ) = A(ω, θ) .

Then settingν := a∗ρ×A∗ρ ∈ Probc(G),

we conclude that the cocycle F(a,A) defined in (3.1) can also be realizedas a cocycle driven by a measure, namely the push forward measureν ∈ Probc(G) above.

The space of mixed cocycles F(a,A) is a metric space with the uniformdistance

dist ((a,A), (a′,A′)) = ‖a− a′‖0 + ‖A −A′‖0 .

Note that the map (a,A) 7→ a∗ρ × A∗ρ ∈ Probc(G) is Lipschitzcontinuous (recall that Probc(G) is equipped with the Wasserstein dis-tance).

3.3. Upper semicontinuity of the Lyapunov exponent. We de-rive a nearly uniform upper semicontinuity of the Lyapunov exponentof a mixed cocycle, a technical result in the spirit of [3, Proposition3.1]. This is a type of uniform upper large deviations estimate, to beemployed in future related projects. For now, as a consequence of thisestimate, we establish the upper semicontinuity of the Lyapunov expo-nent as a function of the measure, relative to the Wasserstein distance.

Fix a number L <∞, let Ω := GL, X = ΩZ and consider the mixedrandom-quasiperiodic dynamics f on X × Td.

Theorem 3.1. Let ν0 ∈ Probc(Ω) be an ergodic measure w.r.t. f .Given any ε > 0, there are δ = δ(ε, ν0, L) > 0, n = n(ε, ν0, L) ∈ N andc = c(ε, ν0, L) > 0 such that for all ν ∈ Probc(Ω) with W1(ν, ν0) < δ,for all θ ∈ Td and for all n ≥ n we have

νZω ∈ X :

1

nlog ‖An(ω)(θ)‖ ≥ L1(ν0) + ε

< e−cn . (3.2)

Moreover, the map ν 7→ L1(ν) is upper semicontinuous with respectto the Wasserstein metric in the space of ergodic measures.

Proof. The argument is similar to the one used in the proof of Theo-rem 2.4. Let

an(ω, θ) := log ‖An(ω)(θ)‖


and note that the sequence ann≥1 is f -subadditive, that is, for alln,m ∈ N and (ω, θ) ∈ X × Td we have:

an+m(ω, θ) ≤ an(ω, θ) + am(fm(ω, θ)) .

For (ω, θ) ∈ X × Td let n(ω, θ) be the least positive integer n suchthat

1

nan(ω, θ) < L1(ν0) + ε . (3.3)

By Kingman’s ergodic theorem, n(ω, θ) is defined for νZ0 × m-a.e.(ω, θ) and, moreover, for m ∈ N, if we denote by

Um := (ω, θ) : n(ω, θ) ≤ m ,

it follows that Um increases to a full νZ0 ×m-measure set as m → ∞.Then there is N = N(ε, ν0) such that νZ0 ×m(UN) < ε.

We note that a priori we do not have an exact analogue of Lemma 2.5,that is, the uniformity in θ of the convergence in Kingman’s ergodictheorem.1 We perform a stopping time argument corresponding to thebehavior of the f -orbit of a point (ω, θ); using the subadditivity of thesequence ann≥1, we eventually reduce the problem to the additivesituation in Theorem 2.4.

Let C = C(L, ν0) := sup

log ‖A(ω0)(θ)‖ : ω0 ∈ Ω, θ ∈ Td<∞.

Fix an arbitrary point (ω, θ) ∈ X × Td and define inductively thesequence of stopping times nk = nk(ω, θ)k≥1 as follows.

If (ω, θ) ∈ UN , let n1 := n(ω, θ), otherwise n1 := 1.If fn1(ω, θ) ∈ UN , let n2 := n(fn1(ω, θ)), otherwise n2 := 1.For k ≥ 1, if fnk+...+n1(ω, θ) ∈ UN then nk+1 := n(fnk+...+n1(ω, θ)),

otherwise nk+1 := 1. Note that for all k ≥ 1 we have 1 ≤ nk ≤ N andby (3.3),

ank(fn1+...+nk−1(ω, θ)) ≤ nk(L1(ν) + ε) + C 1U

N(fn1+...+nk−1(ω, θ)) .

Let n = n(ε, ν,Ω) := N maxCε, 1

, so n ≥ N ≥ n1. Fix any n ≥ n.Since n1 < n1 +n2 < . . . < n1 + . . .+nk < . . ., there is p ≥ 1 such thatn = n1 + . . . np +m, where 0 ≤ m < np+1 ≤ N .

1A posteriori our result provides the upper uniformity in θ. We note that a loweruniformity result, and hence uniform convergence in θ in Kingman’s theorem is ingeneral not possible, see also [4].


Using the subadditivity of the sequence ann≥1 it follows that

an(ω, θ) ≤ an1(ω, θ) + an2(fn1(ω, θ)) + . . .+ anp(f

n1+...+np−1(ω, θ))

+ am(fn1+...+np(ω, θ))

≤ (n1 + . . . np) (L1(ν) + ε) + C

n−1∑j=0

1UN

(f j(ω, θ)) + CN .

Hence for all (ω, θ) ∈ ΩZ × Td and for all n ≥ n we have

1

nlog ‖An(ω)(θ)‖ ≤ L1(ν) + 2ε+ C

1

n

n−1∑j=0

1UN

(f j(ω, θ)) .

The closed set Un ⊂ ΩZ × Td is determined by the coordinatesω0, . . . , ωN−1 and θ. Therefore, as in the proof of Theorem 2.4, us-ing Lemma 2.2, there are an open set D ⊃ UN with (νZ0 ×m) (D) < 2εand a Lipschitz continuous function g : X × Td → [0, 1] which dependonly on the coordinates ω0, . . . , ωN−1, θ such that 1U

N≤ φ ≤ 1D.

Thus for all (ω, θ) ∈ X × Td and n ≥ n we have:

1

n

n−1∑j=0

1UN

(f j(ω, θ)) ≤ 1

n

n−1∑j=0

g(f j(ω, θ)) .

The observable g depends of course only on ε and ν0. ApplyingTheorem 2.4 to g, for any measure ν on Ω that is sufficiently close(depending on ε, ν0) to ν0 in the Wasserstein distance and for all θ ∈ Tdwe have:

1

n

n−1∑j=0

g(f j(ω, θ)) <

∫g d(νZ0 ×m) + ε

for ω outside a set of νZ-measure < e−cn, where c = c(ε, ν0) > 0.Moreover,∫

g d(νZ0 ×m) ≤∫1D d(νZ0 ×m) = (νZ0 ×m)(D) < 2ε,

which combined with the previous estimates proves (3.2).Finally, using the estimate (3.2) and integrating with respect to the

measure νZ×m, where ν is close enough to ν0 in the Wasserstein metric,for all large enough n we have∫

1

nlog ‖An(ω, θ)‖ d(νZ ×m) ≤ L1(ν0) + ε+ Le−cn

< L1(ν0) + 2ε .


Restricting to measures ν that are ergodic with respect to f andletting n→∞ we conclude that L1(ν) < L1(ν0) + 2ε.

Remark 3.2. A uniform lower large deviations estimate (and hence afull, uniform large deviations type estimate) that is, a bound like

νZω ∈ X :

1

nlog ‖An(ω)(θ)‖ ≤ L1(ν0)− ε

< e−cn .

cannot hold at this level of generality.If it did, then (at least restricting to cocycles defined as in Re-

mark 3.1), by the Abstract Continuity Theorem, (see [3, Theorem3.1]) we would conclude that the Lyapunov exponent is a continuousfunction. However, this is not necessarily the case without strongerassumptions on the data.

Indeed, let ν = 12δ(0,I) + 1

2δ(α,A) where (α,A) is the quasiperiodic

cocycle constructed in [12] and shown to be a point of discontinuityof the Lyapunov exponent. Then L1(ν) = 1

2L1(α,A) > 0. However,

ν can be approximated by measures νn with zero Lyapunov exponent,e.g. νn = 1

2δ(0,I) + 1

2δ(α,An), where (α,An)n≥1 is the approximating

sequence of (α,A) in [12]. One may consult [1, Section 5] for moredetails.

4. A motivating example and future work

The study of linear cocycles in general and of mixed cocycles in par-ticular is motivated in part by their relationship with discrete Schrodingeroperators (see [2] for a review of this topic).

Recall the discrete quasiperiodic Schrodinger operator given by

(Hqp(θ)ψ)n = −ψn+1 − ψn−1 + v(θ + nα)ψn, ∀n ∈ Z, (4.1)

for some potential function v ∈ C0(Td,R) and ergodic frequency α ∈Td. Given an energy E ∈ R, consider the corresponding Schrodingercocycle (α, SE), where SE ∈ C0(Td, SL2(R)),

SE(θ) =

(v(θ)− E −1

1 0

).

Let G = Td×C0(Td, SL2(R)) be the space of SL2(R) valued quasiperi-odic cocycles.

We will describe different types of random perturbations of the op-erator (4.1) and the associated mixed Schrodinger cocycle.

Given ρ ∈ Probc(R), consider an i.i.d. sequence of random variableswnn∈Z with common law ρ. Interpreting wnn∈Z as random pertur-bations of the quasiperiodic potential vn(θ) = v(θ+nα), we obtain the


Schrodinger operator

(H ψ)n = −ψn+1 − ψn−1 + (v(θ + nα) + wn) ψn, ∀n ∈ Z. (4.2)

Note that putting P (ω) =

(1 ω0 1

), we can write(

v(θ) + ω − E −11 0

)=

(1 ω0 1

) (v(θ)− E −1

1 0

)= P (ω)SE(θ) .

The Schrodinger cocycle associated to the operator (4.2) is thenthe mixed random-quasiperiodic cocycle driven by the measure νE ∈Probc(G) given by

νE = δα ×∫RδP (ω)SE dρ(ω) . (4.3)

A very different model is obtained if instead we randomize the trans-lation by the frequency α. Given µ ∈ Prob(Td), let αnn∈Z be ani.i.d. sequence of random variables with common law µ and considerthe Schrodinger operator

(H(θ)ψ)n = −ψn+1−ψn−1 +v(θ+α0 + . . .+αn−1)ψn, ∀n ∈ Z. (4.4)

The Schrodinger cocycle associated to the operator (4.4) is the mixedrandom-quasiperiodic cocycle driven by the measure νE ∈ Probc(G)given by

νE = µ× δSE . (4.5)

We may of course randomize both the frequency and the potential,by considering

(H(θ)ψ)n = −ψn+1−ψn−1+(v(θ + α0 + . . .+ αn−1) + wn) ψn, ∀n ∈ Z.The corresponding cocycle is driven by

νE = µ×∫RδP (ω)SE dρ(ω) .

As mentioned before, one of our goals is to study the stability ofthe Lyapunov exponent of a quasiperiodic cocycle under random noise(with appropriate assumptions on the randomness), for Schrodingeror more general cocycles. To this end, in forthcoming papers we willconsider an in depth study of these types of cocycles, as summarizedbelow.

Firstly, we develop results of Furstenberg’s theory on products of ran-dom matrices for our mixed random-quasiperiodic multiplicative pro-cesses. In particular we obtain a Furstenberg-type formula and generic


criteria for the continuity as well as the positivity of the maximal Lya-punov exponent. Under general, easily checkable conditions, these cri-teria are applicable to the mixed Schrodinger cocycles (4.3) and (4.5)thus establishing the continuity and the positivity of the Lyapunov ex-ponents for all energies E. The latter property suggests that in somesense the randomness dominates the quasi-periodicity (under genericassumptions, random multiplicative processes have positive Lyapunovexponents, which is not always the case for quasiperiodic ones). Fur-thermore, it will be interesting to see if, as with the Anderson model,the randomness in the operator (4.2) always leads to Anderson local-ization. This problem will be considered in the future.

The continuity of the Lyapunov exponent mentioned above is noteffective, it is only a qualitative result. We will establish the Holdercontinuity of the Lyapunov exponent of the Schrodinger cocycle (4.2),and in fact for cocycles driven by ν = δα × ρ, where ρ is a measure onC0(Td, SLm(R)). This is obtained via an abstract continuity theorem(ACT) (see [3, Chapter 3]) which depends on the availability of someuniform large deviations type (LDT) estimates on the iterates of thecocycle. The main goal of this future work is deriving such estimates.

We remark that the same problems for cocycles with random fre-quencies such as (4.5), even under stronger regularity assumptions, sofar proved more intractable.

With the above LDT estimate for such mixed random-quasiperiodiccocycles at hand, we will then be able to let the amount of randomnesstend to zero. More precisely, we will establish the stability under ran-dom noise of the LDT estimates for quasiperiodic cocycles. Combinedwith the ACT, this will prove the stability (i.e., in this case, continu-ity) of the Lyapunov exponent of quasiperiodic cocycles under randomperturbations of the cocycle.

Finally, another project will be dedicated to deriving statistical prop-erties for the base mixed random-quasiperiodic dynamics (e.g. largedeviations for more general observables and a central limit theorem).

Acknowledgments. The second author was supported by Fundacaopara a Ciencia e a Tecnologia, under the projects: UID/MAT/04561/2013and PTDC/MAT-PUR/29126/2017. The third author has been sup-ported by the CNPq research grants 306369/2017-6 and 313777/2020-9.


References

[1] L. Backes, Poletti M., and Sanchez A., The set of fiber-bunched cocycles withnonvanishing Lyapunov exponents over a partially hyperbolic map is open,Math. Res. Lett. 25 (2018), no. 6, 1719–1740.

[2] David Damanik, Schrodinger operators with dynamically defined potentials,Ergodic Theory Dynam. Systems 37 (2017), no. 6, 1681–1764.

[3] Pedro Duarte and Silvius Klein, Lyapunov exponents of linear cocycles; con-tinuity via large deviations, Atlantis Studies in Dynamical Systems, vol. 3,Atlantis Press, 2016.

[4] Alex Furman, On the multiplicative ergodic theorem for uniquely ergodic sys-tems, Ann. Inst. H. Poincare Probab. Statist. 33 (1997), no. 6, 797–815.

[5] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math.Statist. 31 (1960), 457–469.

[6] Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc.108 (1963), 377–428.

[7] Michael Hochman, Notes on ergodic theory, (2013).[8] Colin McDiarmid, Concentration, Probabilistic methods for algorithmic dis-

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[9] Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics,vol. 2, Cambridge University Press, Cambridge, 1989, Corrected reprint of the1983 original.

[10] M. Viana, Lectures on Lyapunov exponents, Cambridge Studies in AdvancedMathematics, Cambridge University Press, 2014.

[11] Cedric Villani, Optimal transport, Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], vol. 338,Springer-Verlag, Berlin, 2009, Old and new.

[12] Yiqian Wang and Jiangong You, Examples of discontinuity of Lyapunov ex-ponent in smooth quasiperiodic cocycles, Duke Math. J. 162 (2013), no. 13,2363–2412.

Departamento de Matematica and CMAFcIO, Faculdade de Ciencias,Universidade de Lisboa, Portugal

Email address: [email protected]

Departamento de Matematica and CMAFcIO, Faculdade de Ciencias,Universidade de Lisboa, Portugal


Departamento de Matematica, Pontifıcia Universidade Catolica doRio de Janeiro (PUC-Rio), Brazil


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MIXED RANDOM-QUASIPERIODIC COCYCLES