On Periodic Points of Symplectomorphisms on Surfaces

TL;DR

This paper constructs a specific symplectic flow on higher genus surfaces with minimal fixed points and proves that certain symplectomorphisms isotopic to the identity have infinitely many periodic points under specific conditions, advancing understanding of surface dynamics.

Contribution

It introduces a new symplectic flow with minimal fixed points and establishes conditions for infinite periodic points in symplectomorphisms on higher genus surfaces.

Findings

Constructed a symplectic flow with exactly 2g-2 hyperbolic fixed points and no other periodic orbits.

Proved that symplectomorphisms with a fixed point of non-zero mean index have infinitely many periodic points.

Derived corollaries relating fixed points, elliptic points, and flux conditions to the existence of infinitely many periodic points.

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".