Expansive homeomorphisms of the plane

TL;DR

This paper classifies expansive homeomorphisms of the plane, establishing conditions for conjugacy to linear hyperbolic automorphisms using topological, metric, and Lyapunov function techniques.

Contribution

It provides necessary and sufficient conditions for such homeomorphisms to be conjugate to linear hyperbolic automorphisms, extending results to non-compact settings.

Findings

Characterization of expansive homeomorphisms via Lyapunov metrics

Generalization of compact results to the plane at infinity

New classification theorem based on local properties

Abstract

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.