Periodic approximation of Lyapunov exponents for Banach cocycles

TL;DR

This paper establishes a method to approximate Lyapunov exponents of Banach cocycles over hyperbolic systems using periodic orbits, providing new insights into their growth and distortion properties.

Contribution

It introduces a periodic approximation technique for Lyapunov exponents of Banach cocycles and constructs a Lyapunov norm in infinite dimensions.

Findings

Lyapunov exponents can be approximated by periodic orbit data

Upper and lower exponents may not always be approximable by periodic measures

Provides estimates for cocycle growth and quasiconformal distortion

Abstract

We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism $f$ of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is H\"older continuous. We prove that upper and lower Lyapunov exponents of $A$ with respect to an ergodic invariant measure $\mu$ can be approximated in terms of the norms of the values of $A$ on periodic orbits of $f$. We also show that these exponents cannot always be approximated by the exponents of $A$ with respect to measures on periodic orbits. Our arguments include a result of independent interest on construction and properties of a Lyapunov norm for infinite dimensional setting. As a corollary, we obtain estimates of the growth of the norm and of the quasiconformal distortion of the cocycle in terms of the growth at the periodic points of $f$.