Volume 8 (2025)
Guihéneuf, Pierre-Antoine;  Le Calvez, Patrice;  Passeggi, Alejandro
Area preserving homeomorphisms of surfaces with rational rotational direction
Annales Henri Lebesgue, Volume 8 (2025), pp. 329-372.

Metadata

DOI10.5802/ahl.237
Keywords Rotation vector, maximal isotopy, transverse foliation

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 $\operatorname{rot}_f(\lambda )\in H_1(S,\mathbb{R})$ 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.


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