# Lyapunov exponents for expansive homeomorphisms

## TL;DR

This paper introduces Lyapunov exponents for expansive homeomorphisms on compact metric spaces, establishing conditions under which these exponents indicate hyperbolicity and attractor properties, extending classical concepts to more general spaces.

## Contribution

It defines Lyapunov exponents for expansive homeomorphisms on compact metric spaces and proves their significance in hyperbolicity and attractor characterization, especially on Peano spaces.

## Key findings

- Lyapunov exponents are positive and negative under certain conditions on Peano spaces.
- Negative maximal Lyapunov exponent implies the invariant set is an attractor.
- Exponents coincide with classical ones for diffeomorphisms on manifolds.

## Abstract

Let (M,d) be a compact metric space and f:M --> M an expansive homeomorphism. We define Lyapunov exponents L(f,m)_{max} and l(f,mu)_{min} for an f-invariant measure m. When L(f,m)_{max} > 0 and l(f,mu)_{min} < 0 can be interpreted as a weak form of hyperbolicity for f. We prove that if M is a Peano space then there is g>0 such that L(f,m)_{max} > g and l(f,m)_{min}< - g. We also show that the hypothesis that M is a Peano space is necessary to obtain the maximal Lyapunov exponent positive and the minimal Lyapunov exponent negative. Moreover we define Lyapunov exponents for K, a compact f-invariant subset of M and prove that if the maximal Lyapunov exponent of K is negative then K is an attractor. When f is a diffeomorphism on a compact manifold, these Lyapunov exponents coincide with the usual ones.