Let S be a closed surface of genus g≥1, furnished with an area form ω. We show that there exists an open and dense set Or
Nearby in the stack
of the space of Hamiltonian diffeomorphisms of class
Cr
,
1≤r≤∞
, endowed with the
Cr
-topology, such that every
f∈Or
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].