#### Vol. 294, No. 1, 2018

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On periodic points of symplectomorphisms on surfaces

### Marta Batoréo

Vol. 294 (2018), No. 1, 19–40
##### 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