On Periodic Points of Symplectomorphisms on Surfaces

Abstract

We construct a symplectic flow on a surface of genus g greater than one with exactly 2g-2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly non-degenerate) symplectomorphism of a surface (with genus g greater than one) isotopic to the identity has infinitely many periodic points if there exists a fixed point with non-zero mean index. From this result, we obtain two corollaries, namely that such a symplectomorphism 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".

21 pages, 7 figures

References in corpus (7)

• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms
• A symplectic proof of a theorem of Franks
• Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity
• On hyperbolic points and periodic orbits of symplectomorphisms
• Non-contractible periodic trajectories of symplectic vector fields, Floer cohomology and symplectic torsion
• On non-contractible hyperbolic periodic orbits and periodic points of symplectomorphisms

Cited by in corpus (2)

• Periodic points of rational area-preserving homeomorphisms
• Invariant Sets and Hyperbolic Closed Reeb Orbits