Related papers: Furstenberg Theory of Mixed Random-Quasiperiodic C…

It is known that the Lyapunov exponent for multifrequency analytic cocycles is weak-H\"older continuous in cocycle for certain Diophantine frequencies, and that this implies certain regularity of the integrated density of states in energy…

We give a criterion implying subcritical behavior for quasi-periodic Schr\"odinger operators where the potential sampling function is given by a trigonometric polynomial. Subcritical behavior, in the sense of Avila's global theory, is known…

We study the quantitative simplicity of the Lyapunov spectrum of $d$-dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive…

A theorem of Viana says that almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. In this note we extend this result to cocycles on any noncompact classical semisimple Lie group.

It is shown that the asymptotic spectra of finite-time Lyapunov exponents of a variety of fully chaotic dynamical systems can be understood in terms of a statistical analysis. Using random matrix theory we derive numerical and in particular…

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.

We prove spectral and dynamical localization for a one-dimensional Dirac operator to which is added an ergodic random potential, with a discussion on the different types of potential. We use scattering properties to prove the positivity of…

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on…

It is classical, following Furstenberg's theorem on positive Lyapunov exponent for products of random SL$(2, \mathbb R)$ matrices, that the one dimensional random Schr\"odinger operator has Anderson localization at arbitrary disorder. This…

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…

We observe that under certain conditions on the Lyapunov exponents a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a H\"{o}lder continuous map $A:M\to M(d, \mathbb{R})$ over a…

Enhancing a recent result of Bayart and Ruzsa we obtain a Birkhoff-type characterization of upper frequently hypercyclic operators and a corresponding Upper Frequent Hypercyclicity Criterion. As an application we characterize upper…

For analytic quasi-periodic Schr\"odinger cocycles, You and Zhang [9] proved that the Lyapunov exponent is H\"older continuous for weak Liouville frequency. In this paper, we prove that the H\"older continuity also holds if the analytic…

In this work, we are interested in the study of the upper Lyapunov exponent $\lambda^+(\theta)$ associated to the periodic family of cocycles defined by $$A_\theta(x):=A(x)R_\theta,\qquad x\in X,$$ where $A\::\: X\to…

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are H\"older continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that…

Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schroedinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical…

We consider discrete one-dimensional Schr\"odinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant H\"older…

In this paper we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schr\"odinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov…

We prove that for semi-invertible and H\"older continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with…

We prove that the Lyapunov exponents of typical fiber bunched linear cocycles over Lorenz-like flows have multiplicity one: the set of exceptional cocycles has infinite codimention, i.e. it is locally contained in finite unions of closed…