Area preserving homeomorphisms of surfaces with rational rotational direction

Let S be a closed surface of genus g≥2, furnished with a Borel probability measure λ with total support. We show that if f is a λ-preserving homeomorphism isotopic to the identity such that the rotation vector rot f (λ)∈H 1 (S,ℝ) is a multiple of an element of H 1 (S,ℤ), then f has infinitely many periodic orbits. Moreover, these periodic orbits can be supposed to have their rotation vectors arbitrarily close to the rotation vector of any fixed ergodic Borel probability measure.