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

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.