On non-contractible periodic orbits for surface homeomorphisms

TL;DR

This paper investigates conditions under which surface homeomorphisms homotopic to the identity have non-contractible periodic orbits, providing a trichotomy that characterizes their dynamical behavior.

Contribution

It establishes a new trichotomy for such homeomorphisms, linking fixed points, periodic points, and orbit bounds, with implications for rotation set analysis.

Findings

Either the fixed point set projects to an essential continuum

Existence of non-contractible periodic points of arbitrarily large period

Existence of a uniform bound on the diameter of contractible periodic orbits

Abstract

In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if $\hat g$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) The set of fixed points for $\hat g$ projects to a closed subset $F$ which contains an essential continuum, (2) $g$ has non-contratible periodic points of every sufficiently large period, or (3) there exists an uniform bound $M$ such that, if $\hat x$ projects to a contractible periodic point then the $\hat g$ orbit of $\hat x$ has diameter less or equal to $M$. Some consequences for homeomorphisms of surfaces whose rotation set is a singleton are derived.