Related papers: An invitation to $\text{SL}_2(\mathbb{R})$ cocycle…
Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron-Frobenius operators associated to the maps. Of…
We show that for a large class of potential functions and big coupling constant $\lambda$ the Schr\"odinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$ for all…
It is given notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic…
This is the second part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general…
In this paper we use a path-integral approach to represent the Lyapunov exponents of both deterministic and stochastic dynamical systems. In both cases the relevant correlation functions are obtained from a (one-dimensional) supersymmetric…
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…
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…
In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some…
The paper is devoted to the properties of a complex matrix ``twisted,'' otherwise called ``spectral,'' cocycle, associated with substitution dynamical systems. Following a recent finding of Rajabzadeh and Safaee [arXiv:2501.16824] of an…
In this work we study relations between regularity of invariant foliations and Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of desintegration of Lebesgue measure…
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…
This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory,…
In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $\Phi:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear…
We prove measurable Livsic theorems for dynamical systems modelled by Markov Towers. Our regularity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We…
Lyapunov exponents are indicators for the chaotic properties of a classical dynamical system. They are most naturally defined in terms of the time evolution of a set of so-called covariant vectors, co-moving with the linearized flow in…
We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. This builds on work of Guivarc'h-Raugi and Gol'dsheid-Margulis, who considered products of random matrices,…
We survey a collection of recent results on center Lyapunov exponents of partially hyperbolic diffeomorphisms. We explain several ideas in simplified setups and formulate the general versions of results. We also pose some open questions.
In this paper, we present a class of random Schr\"odinger cocycles showing that, for random cocycles with non-compact support, the presence of certain finite moment conditions is essential for establishing a specific modulus of continuity…
We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.
We consider continuous $SL(2,R)$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.