On the struture of homeomorphims of the open annulus

TL;DR

This paper investigates the structure of certain homeomorphisms of the open annulus, showing they can be conjugated to translations on dense open sets and analyzing their action on specific components.

Contribution

It provides a detailed analysis of homeomorphisms of the open annulus, demonstrating conjugacy to translations on dense open sets and examining their action on connected components.

Findings

Existence of an invariant dense open set where the homeomorphism is conjugate to a translation.

Characterization of the homeomorphism's action on compactly connected components.

Insights into the structure of homeomorphisms without fixed points and wandering points.

Abstract

Let $h$ be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that $h$ admits a $h$-invariant dense open set $O$ on which it is conjugate to a translation and we study the action of $h$ on the compactly connected components of the closed and without interior set ${\bf R}^2 \setminus O$.