2016/12/26 by Conejeros, Jonathan, Tal, Fabio Armando
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1612.08439
In this article we consider homeomorphisms of the open annulus \mathbbA=ℝ/ℤ× 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 \checkf a lift of f to the universal covering of \mathbbA such that the set of fixed points of \checkf is non-empty and that this set projects into an open topological disk of \mathbbA. 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 \mathbbA there exists a constant M (depending on the compact set) such that, if \checkz and \checkfn(\checkz) 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.