Homologically non-trivial periodic orbits for generic Hamiltonian diffeomorphisms of surfaces

Let $ S $ be an oriented closed surface of genus $ g\geq 1 $, furnished with an area form $ \omega $. We show that for $ 1\leq r\leq \infty $, there exists an open and dense set $ {\mathcal O_r} $ of the space of Hamiltonian diffeomorphisms of class $ C^r $, endowed with the $ C^r $-topology, such that every $ f\in \mathcal O_r $ possesses infinitely many periodic orbits with nonzero rotation vector. Similar results hold if one replaces the space of Hamiltonian diffeomorphisms with the space of symplectic diffeomorphisms that are isotopic to the identity. Moreover, we give some details about the possible homological directions. In the Hamiltonian case, we obtain a positive answer to a question asked by Viktor Ginzburg and Başak Gürel concerning existence of non-contractible periodic orbits. The proof is a consequence of recent previous works of the authors [15].