Area preserving homeomorphisms of surfaces with rational rotational direction

Abstract

Let $S$ be a closed surface of genus $g\ge 2$, 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.

38 pages, 6 figures