Area preserving homeomorphisms of surfaces with rational rotational direction

Abstract

Let $S$ be a closed surface of genus $g\geq 2$, furnished with a Borel probability measure $\lambda$ with total support. We show that if $f$ is a $\lambda$-preserving homeomorphism isotopic to the identity such that the rotation vector $\mathrm{rot}_f(\lambda)\in H_1(S,\mathbb R)$ is a multiple of an element of $H_1(S,\mathbb Z)$, 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.