Existence of Non-Contractible Periodic Orbits for Homeomorphisms of the Open Annulus

TL;DR

This paper proves that certain homeomorphisms of the open annulus either have non-contractible periodic points of arbitrarily large prime period or exhibit bounded behavior in their lifts, with implications for rotation sets.

Contribution

It establishes a dichotomy for homeomorphisms of the open annulus with specific fixed point properties, linking the existence of non-contractible periodic orbits to boundedness conditions.

Findings

Existence of non-contractible periodic points of arbitrarily large prime period under certain conditions.

Boundedness condition on the projections of lifted points for all compact sets.

Implications for homeomorphisms with rotation set reduced to an integer.

Abstract

In this article we consider homeomorphisms of the open annulus $\mathbb{A}=\mathbb{R}/\mathbb{Z}\times \mathbb{R}$ which are isotopic to the identity and preserve a Borel probability measure of full support, focusing on the existence of non-contractible periodic orbits. Assume $f$ such homeomorphism such that the connected components of the set of fixed points of $f$ are all compact. Further assume that there exists $\check{f}$ a lift of $f$ to the universal covering of $\mathbb{A}$ such that the set of fixed points of $\check{f}$ is non-empty and that this set projects into an open topological disk of $\mathbb{A}$. We prove that, in this setting, one of the following two conditions must be satisfied: (1) $f$ has non-contractible periodic points of arbitrarily large prime period, or (2) for every compact set $K$ of $\mathbb{A}$ there exists a constant $M$ (depending on the compact set) such that, if $\check{z}$ and $\check{f}^n(\check{z})$ project on $K$, then their projections on the first coordinate have distance less or equal to $M$. Some consequence for homeomorphisms of the open annulus whose rotation set is reduced to an integer number are derived.