  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].
