We derive a criterion for the positivity of the maximal Lyapunov exponent of generic mixed random-quasiperiodic linear cocycles, a model introduced in a previous work. This result is applicable to cocycles corresponding to Schrödinger operators with randomly perturbed quasiperiodic potentials. Moreover, we establish an average uniform convergence to the Lyapunov exponent in the Oseledets theorem.
We introduce the concept of mixed random-quasiperiodic linear cocycles. We characterize the ergodicity of the base dynamics and establish a large deviations type estimate for certain types of observables. For the fiber dynamics we prove the uniform upper semicontinuity of the maximal Lyapunov exponent. This paper is meant to introduce a model to be studied in depth in further projects.
It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.
This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint Hölder continuity of the maximal Lyapunov exponent as a function of the cocycle and the transition kernel in the vicinity of any irreducible cocycle with simple maximal Lyapunov exponent. Our approach, via Furstenberg's formula, shows the Hölder continuous dependence on the data of the stationary measure of the projective cocycle and in particular provides a more computable Hölder exponent.
We consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of ba…
Pointwise modulus of continuity of the Lyapunov exponent and integrated density of states for analytic multi-frequency quasiperiodic $M(2, \mathbb{C})$ cocycles
We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space $G^{s}$ with $s<2$ \cite{klein,cgyz}. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.
We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.
For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the "Schrödinger conjecture" and the "Last's intersection spectrum conjecture" have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open pr…
Given a quasiperiodic cocycle in sl(2, R) sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the H{ö}lder regularity of the Lyapunov exponent.
Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,C)-reducible is in fact G-reducible modulo 2 if G=GL(n,R),SL(n,R),Sp(n,R) or O(n) and modulo 1 if G=U(n).
We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renorm…
Sarnak's Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics. We prove that Möbius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extend \cite{LS15,W17} to the noncommutative case. The proof relies on quantitative version of almost reducibility.
Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d} \times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in PDEs.
In this paper we first obtain a formula of averaged Lyapunov exponents for ergodic Szego cocycles via the Herman-Avila-Bochi formula. Then using acceleration, we construct a class of analytic quasi-periodic Szego cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in [Y] allows us to estimate the Lebesgue measure of support of the measure associated to certain class of C1 quasiperiodic 2- sided Verblunsky coefficients. Using the same method, we also recover the [S-S] results for Schrodinger cocycles with nonconstant real analytic potentials an…
We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schrödinger cocycles are uniformly positive and weak Hölder continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak Hölder continous. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general $\mathrm{SL}(2,\mathbb R)$ cocycles, which in turn can be applied to get uniform positivit…
We continue our study of the local theory for quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $G=SU(2)$, over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the space orthogonal to $L^{2}(\mathbb{T} ^{d}) \hookrightarrow L^{2}(\mathbb{T} ^{d} \times G)$. Subsequently, we describe the equivalence classes of cocycles under smooth conjugacy, as a function of the parameters defining their K.A.M. normal form. Fina…
In this mémoire we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty}$ reducible cocycles are dense in the $C^{\infty}$ topology, for a full measure set of frequencies. Moreover, we will show that every cocycle (or an appropriate iterate of it, if homotopy appears as an obstruction) is almost torus-reducible (i.e. can be conjugated arbitrarily close to cocycles taking values in an abelian subgroup of G). In the course of the proof we will firstly define two invarian…
Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Di…
The celebrated Oseledets theorem \cite{O}, building over seminal works of Furstenberg and Kesten on random products of matrices and random variables taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg}, ensures that the Lyapunov exponents of $\mathrm{SL}(d,\mathbb R)$-cocycles $(d\geqslant 2)$ over the shift are well defined for all points in a total probability set, ie, a full measure subset for all invariant probabilities. Given a locally constant $\mathrm{SL}(d,\mathbb R)$-valued cocycle we are interested both in the set of points on the shift space for which some Lyapun…
We study the linear cohomological equation in the smooth category over quasi-periodic cocycles in $\mathbb{T} ^{d} \times SU(2)$. We prove that, under a full measure condition on the rotation in $\mathbb{T} ^{d}$, for a generic cocycle in an open set of cocycles, the equation admits a solution for a dense set of functions on $\mathbb{T} ^{d} \times SU(2)$ of zero average with respect to the Haar measure. This property is known as Distributional Unique Ergodicity (DUE). We then show that given such a cocycle, for a generic function no such solution exists. We thus confirm in this context a conj…
Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarc'h-Raugi, Gol'dsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main …
For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimension central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variatio…
We derive a series summation formula for the average logarithm norm of the action of a matrix on the projective space. This formula is shown to be useful to evaluate some Lyapunov exponents of random $\SL$-matrix cocycles, which include a special class for which H. Furstenberg had provided an explicit integral formula.
A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier and generalized to the context of non-linear cocycles by Avila and Viana, gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and in particular, obtain a simpler proof for some of the known invariance principle results. As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose cente…
In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmueller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmueller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmueller curves. Fina…
