On periodic points of symplectomorphisms on surfaces
Copyright policy )On periodic points of symplectomorphisms on surfaces
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
Symplectic mappings, fixed points (dynamical systems), symplectomorphism, Mathematics - Symplectic Geometry, Symplectic aspects of Floer homology and cohomology, 53D40 (primary), 37J10, 70H12 (secondary), Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics, FOS: Mathematics, Symplectic Geometry (math.SG), Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Floer homology
Symplectic mappings, fixed points (dynamical systems), symplectomorphism, Mathematics - Symplectic Geometry, Symplectic aspects of Floer homology and cohomology, 53D40 (primary), 37J10, 70H12 (secondary), Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics, FOS: Mathematics, Symplectic Geometry (math.SG), Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Floer homology
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