Related papers: Random 2D linear cocycles II: statistical properti…
We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value…
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…
The theory of products of random matrices and Lyapunov exponents have been widely studied and applied in the fields of biology, dynamical systems, economics, engineering and statistical physics. We consider the product of an i.i.d. sequence…
For an open and dense subset of elliptic ${\rm SL}(2,\mathbb R)$ matrix cocycles, we construct a family of loosely Bernoulli ergodic measures with zero top Lyapunov exponent. This provides a counterpart to a classical result by Furstenberg.…
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 derive large deviations type (LDT) estimates for linear cocycles over an ergodic multifrequency torus translation. These models are called quasi-periodic cocycles. We make the following assumptions on the model: the translation vector…
It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In…
A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system…
We prove a formula for the speed of distance stationary random sequences. A particular case is the classical formula for the largest Lyapunov exponent of an i.i.d. product of two by two matrices in terms of a stationary measure on…
In this paper we prove the continuity of all Lyapunov exponents, as well as the continuity of the Oseledets decomposition, for a class of irreducible cocycles over strongly mixing Markov shifts. Moreover, gaps in the Lyapunov spectrum lead…
We establish some conditions under which $\text{GL}(d,\mathbb{R})$-valued cocycles over a subshift of finite type, equipped with an equilibrium state, exhibit exponential asymptotics for the spectral radius. Specifically, we show that the…
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $\mu$. The cocycle $A$ over $f$ is Holder…
We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential…
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…
We prove a new characterization of uniform hyperbolicity for fiber-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched $SL(2,\mathbb{R})$-cocycle defined over a…
We develop a higher-dimensional extension of multifractal analysis for typical fiber-bunched linear cocycles. Our main result is a relative variational principle, which shows that the topological entropy of Lyapunov exponent level sets can…
We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the…
We give a new proof of E. Le Page's theorem on the Holder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first Lyapunov exponent, as well as the…
In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov expoent for typical matrix cocyles. By using the results we obtain, we prove that in any…
This paper is concerned with the Lyapunov spectrum for measurable cocycles over an ergodic pmp system taking values in semi-simple real Lie groups. We prove simplicity of the Lyapunov spectrum and its continuity under certain perturbations…