Periodic approximation of exceptional Lyapunov exponents for semi-invertible operator cocycles

TL;DR

This paper demonstrates that for certain infinite-dimensional dynamical systems, exceptional Lyapunov exponents can be approximated by those computed on periodic orbits, extending finite-dimensional results.

Contribution

It establishes a method to approximate exceptional Lyapunov exponents in infinite-dimensional systems using periodic orbit measures, under broad conditions.

Findings

Approximation of Lyapunov exponents on periodic orbits

Applicability to a wide class of infinite-dimensional systems

Extension of finite-dimensional approximation results

Abstract

We prove that for semi-invertible and H\"older continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.