Linearization of topologically Anosov homeomorphisms of non compact surfaces

TL;DR

This paper classifies topologically Anosov homeomorphisms on non-compact genus-zero surfaces, showing they are conjugate to homotheties or reverse homotheties on the plane, confirming a conjecture for this class.

Contribution

It proves that such homeomorphisms on non-compact genus-zero surfaces are conjugate to simple scaling maps, extending previous conjectures.

Findings

S=R^2 for these homeomorphisms

f is conjugate to a homothety or reverse homothety

Weaker conjecture confirmed for genus-zero surfaces

Abstract

We study the dynamics of Topologically Anosov homeomorphisms of non compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f:S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \R ^ 2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in a previous work.