Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

Boris Kalinin; Victoria Sadovskaya

doi:10.3934/dcds.2018224

Article Contents

2018, Volume 38, Issue 10: 5105-5118. Doi: 10.3934/dcds.2018224

This issue Previous Article Characterization of noncorrelated pattern sequences and correlation dimensions Next Article Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

Boris Kalinin and
Victoria Sadovskaya

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: December 2017

Published: October 2018

The first author was supported in part by Simons Foundation grant 426243, the second author was supported in part by NSF grant DMS-1301693.

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Abstract

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 $μ$. The cocycle $\mathcal{A}$ over $f$ is Hölder continuous and takes values in $GL(d, \mathbb{R})$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d, \mathbb{R})$-valued cocycle $\mathcal{A}$ we prove that the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the Lyapunov exponents of $\mathcal{A}$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $\mathcal{A}$ with respect to $μ$, but they cannot always be approximated by the exponents of $\mathcal{A}$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $\mathcal{A}$ on hyperbolic periodic orbits of $f$.

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Mathematics Subject Classification: Primary: 37H15, 37D25.

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