On periodic points of symplectomorphisms on surfaces

Marta Batoréo

Vol. 294 (2018), No. 1, 19–40

DOI: 10.2140/pjm.2018.294.19

Abstract

We construct a symplectic flow on a surface of genus $g\ge 2$, ${\Sigma }_{g\ge 2}$, with exactly $2g-2$ hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly nondegenerate) symplectomorphism of ${\Sigma }_{g\ge 2}$ isotopic to the identity has infinitely many periodic points if there exists a fixed point with nonzero mean index. From this result, we obtain two corollaries, namely that such a symplectomorphism of ${\Sigma }_{g\ge 2}$ with an elliptic fixed point or with strictly more than $2g-2$ fixed points has infinitely many periodic points provided that the flux of the isotopy is “irrational”.

Keywords

symplectomorphisms, surfaces, Floer homology

Mathematical Subject Classification 2010

Primary: 53D40

Secondary: 37J10, 70H12

Milestones

Received: 9 March 2017

Revised: 11 September 2017

Accepted: 20 September 2017

Published: 5 January 2018