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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 ${\mathcal O_r}$ ofthe space of Hamiltonian diffeomorphisms of class $C^r$, $1\leq r\leq\infty$,endowed with the $C^r$-topology, such that every $f\in \mathcal O_r$ possessesinfinitely many non contractible periodic orbits. We obtain a positive answerto a question asked by Viktor Ginzburg and Ba\c{s}ak G\"{u}rel. The proof is aconsequence of recent previous works of the authors [LecSa].

Non contractible periodic orbits for generic hamiltonian diffeomorphisms of surfaces