Non contractible periodic orbits for generic hamiltonian diffeomorphisms of surfaces

TL;DR

This paper proves that for generic Hamiltonian diffeomorphisms on surfaces of genus at least one, there are infinitely many non-contractible periodic orbits, answering a question posed by Ginzburg and G"urel.

Contribution

It establishes the existence of infinitely many non-contractible periodic orbits for a generic set of Hamiltonian diffeomorphisms on surfaces, extending previous results.

Findings

Generic Hamiltonian diffeomorphisms have infinitely many non-contractible periodic orbits.

The result applies to a dense and open subset in the $C^r$ topology.

The proof builds on recent work by the authors.

Abstract

Let $S$ be a closed surface of genus $g\geq 1$, furnished with an area form $\omega$. We show that there exists an open and dense set ${\mathcal O_r}$ of the space of Hamiltonian diffeomorphisms of class $C^r$, $1\leq r\leq\infty$, endowed with the $C^r$-topology, such that every $f\in \mathcal O_r$ possesses infinitely many non contractible periodic orbits. We obtain a positive answer to a question asked by Viktor Ginzburg and Ba\c{s}ak G\"{u}rel. The proof is a consequence of recent previous works of the authors [LecSa].