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

We construct a symplectic flow on a surface of genus g 2, Σg2, with exactly 2g 2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly nondegenerate) symplectomorphism of Σg2 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 Σg2 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”.

symplectomorphisms, surfaces, Floer homology
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 37J10, 70H12
Received: 9 March 2017
Revised: 11 September 2017
Accepted: 20 September 2017
Published: 5 January 2018
Marta Batoréo
Departamento de Matemática
Universidade Federal do Espírito Santo
Campus de Goiabeiras