Non contractible periodic orbits for generic hamiltonian diffeomorphisms of surfaces

Let $S$ be a closed surface of genus $g\ge 1$, 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 $C^r$, $1\le r\le \mathrm{\infty }$, 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 Başak Gürel. The proof is a consequence of recent previous works of the authors [LecSa].