On non-contractible periodic orbits for surface homeomorphisms

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.