Conservative surface homeomorphisms with finitely many periodic points

The goal of the article is to characterize the conservative homeomorphisms of a closed orientable surface $S$ of genus $\ge 2$, that have finitely many periodic points. By conservative, we mean a map with no wandering point. As a particular case, when $S$ is furnished with a symplectic form, we characterize the symplectic diffeomorphisms of $S$ with finitely many periodic points.