The number of periodic points of surface symplectic diffeomorphisms

TL;DR

This paper proves that symplectic diffeomorphisms on surfaces with many fixed points have infinitely many periodic points, and constructs examples with specific periodic point counts, including flows with only one fixed point.

Contribution

It establishes a smooth variant of Franks theorem for surfaces and provides explicit examples of symplectic flows with prescribed periodic point properties.

Findings

More than 2g-2 fixed points imply infinitely many periodic points.

Constructed symplectic flows with only one fixed point and no other periodic orbits.

Demonstrated the existence of symplectic diffeomorphisms with a prescribed number of periodic points.

Abstract

We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus $g$; it implies that a symplectic diffeomorphism isotopic to the identity with more than $2g-2$ fixed points, counted homologically, has infinitely many periodic points. Furthermore, we present examples of symplectic diffeomorphisms with a prescribed number of periodic points. In particular, we construct symplectic flows on surfaces possessing only one fixed point and no other periodic orbits.