Dense area-preserving homeomorphisms have zero Lyapunov exponents

TL;DR

This paper introduces a new Lyapunov exponent concept for continuous maps, showing that generic area-preserving homeomorphisms on surfaces have zero exponents almost everywhere, and discusses the topological properties of this exponent.

Contribution

It defines a new Lyapunov exponent for continuous maps, proves its equivalence to classical exponents for differentiable maps, and analyzes its generic behavior and topological properties in surface homeomorphisms.

Findings

Generic area-preserving homeomorphisms have zero Lyapunov exponents almost everywhere.

The new Lyapunov exponent coincides with classical exponents for differentiable maps.

The integral of the top Lyapunov exponent is not upper-semicontinuous in the C0-topology.

Abstract

We give a new definition for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a C0-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the C0-topology, to the integral (with respect to area) of its top new Lyapunov exponent over the whole surface cannot be upper-semicontinuous.