Conservative surface homeomorphisms with finitely many periodic points

TL;DR

This paper characterizes conservative surface homeomorphisms with finitely many periodic points, focusing on maps with no wandering points on closed orientable surfaces of genus at least 2, including symplectic diffeomorphisms.

Contribution

It provides a characterization of conservative homeomorphisms with finitely many periodic points on higher genus surfaces, extending understanding of their dynamical properties.

Findings

Characterization of conservative homeomorphisms with finitely many periodic points.

Extension of results to symplectic diffeomorphisms on surfaces.

Insights into the structure of maps with limited periodic behavior.

Abstract

The goal of the article is to characterize the conservative homeomorphisms of a closed orientable surface $S$ of genus $\geq 2$, 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.