On Non-contractible Periodic Orbits and Bounded Deviations

TL;DR

This paper establishes a dichotomy for surface homeomorphisms near the identity, linking bounded deviations in the universal cover to the existence of non-contractible periodic points, and characterizes dynamics of area-preserving torus maps.

Contribution

It introduces a new dichotomy for surface homeomorphisms, connecting bounded deviations and non-contractible periodic points, and applies this to classify area-preserving torus dynamics.

Findings

Either orbits are uniformly bounded in the universal cover or non-contractible periodic points exist.

For area-preserving torus maps without non-contractible periodic points, dynamics are either bounded or have a strong irrational direction.

Abstract

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic points. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic points, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or the lifted map has a single strong irrational dynamical direction.