Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems

TL;DR

This paper demonstrates that for certain hyperbolic dynamical systems and Banach cocycles, one can approximate measure-theoretic entropy and Lyapunov exponents using horseshoes and topological entropy.

Contribution

It establishes a method to approximate Lyapunov exponents and entropy for Banach cocycles over nonuniformly hyperbolic systems via horseshoes and dominated splittings.

Findings

Existence of horseshoes approximating entropy and Lyapunov exponents.

Approximation of measure-theoretic entropy by topological entropy.

Approximation of Lyapunov exponents on horseshoes.

Abstract

Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $\mu$ with positive entropy, and let $\mathcal{A}$ be a H\"older continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $\mu$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mathcal{A}$ on the horseshoes, respectively.