Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

TL;DR

This paper studies Lyapunov exponents of linear cocycles over non-uniformly hyperbolic systems, showing approximation results for finite-dimensional cases and bounds for infinite-dimensional cases, advancing understanding of stability in complex dynamical systems.

Contribution

It proves that Lyapunov exponents for finite-dimensional cocycles can be approximated by those on hyperbolic periodic orbits, and provides bounds for infinite-dimensional cocycles.

Findings

Lyapunov exponents for finite-dimensional cocycles are approximable by periodic orbit exponents.

Upper and lower Lyapunov exponents in infinite-dimensional cases can be approximated by return values on periodic orbits.

The results extend understanding of stability and spectral properties in non-uniformly hyperbolic systems.

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 $\mu$. The cocycle $A$ over $f$ is Holder continuous and takes values in $GL(d,R)$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d,R)$-valued cocycle $A$ we prove that the Lyapunov exponents of $A$ with respect to $\mu$ can be approximated by the Lyapunov exponents of $A$ 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 $A$ with respect to $\mu$, but they cannot always be approximated by the exponents of $A$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $A$ on hyperbolic periodic orbits of $f$.