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.

Minor changes. To appear in Ergodic Theory and Dynamical Systems

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• Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion
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Cited by in corpus (1)

• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds