Related papers: Furstenberg Theory of Mixed Random-Quasiperiodic C…
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…
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 H\"older continuity of the maximal Lyapunov exponent as a function of the cocycle and the…
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…
An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given…
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…
We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov…
Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such…
We study lower bounds on the Lyapunov exponent associated with one-frequency quasiperiodic Schr\"odinger operators with an added finite valued background potential. We prove that, for sufficiently large coupling constant, the Lyapunov…
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…
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…
This paper serves as an extended road map for our long-term project "Mixed Random-quasiperiodic Cocycles" [arXiv:2201.04745, arXiv:2109.09544, arXiv:2210.16908, 6, 7] with Pedro Duarte and Silvius Klein. Despite exhibiting totally different…
We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schr\"odinger cocycles are uniformly positive and weak H\"older continuous as function of energies. As…
For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive…
We study multi-frequency quasiperiodic Schr\"{o}dinger operators on $\mathbb{Z} $. We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of…
We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by…
We consider one-dimensional quasi-periodic Schr\"odinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal H\"older continuity of the Lyapunov exponents…
We consider Holder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we…
We study the problem of estimating the maximal Lyapunov exponent of dominated cocycles. In particular we are concerned with cocycles over Gibbs states on shifts of finite type for which both the function defining the cocycle and the…
The Lyapunov exponent characterizes the asymptotic behavior of long matrix products. Recognizing scenarios where the Lyapunov exponent is strictly positive is a fundamental challenge that is relevant in many applications. In this work we…
This paper studies structured products of real matrices for which the top Lyapunov exponent can be accessed by reducing the dynamics to an amenable generalization of upper triangular matrices. Exploiting prescribed zero patterns (including…