CONSERVATIVE SURFACE HOMEOMORPHISMS WITH FINITELY MANY PERIODIC POINTS

We give a characterization of homeomorphisms $f$ of a closed surface of genus $\geq 2$ with no wandering point that have finitely many periodic points. The main result is the fact that there exists an integer $q$ such that the periodic points of $f^q$ are fixed and $f^q$ is isotopic to the identity relative to its fixed point set. The emblematic way to construct such an example is to start with the time one map of a flow of minimal direction for a translation surface, to add finitely many stopping points and to lift this map to a finite covering. The main result in the proof is that every homeomorphism with no wandering point, isotopic to a Dehn twist map, has infinitely many periodic points. Such a result was known for a generic area preserving diffeomorphism in the isotopy class. To extend this result, obtained with Martin Sambarino, to the general case, one needs to introduce a ``forcing lemma'' , very similar to a forcing result obtained with Fabio Tal in the case of maps isotopic to the identity.