Refinements of Franks' theorem and applications in Reeb dynamics

TL;DR

This paper refines Franks' theorem to establish the existence and symmetry of periodic orbits in annulus homeomorphisms, with applications to Reeb dynamics and celestial mechanics.

Contribution

It provides new refinements of Franks' theorem that include symmetry considerations and applies these to Reeb dynamics and celestial mechanics.

Findings

Existence of infinitely many periodic orbits with periods prime to a given number.

Symmetric periodic orbits in reversible systems.

Applications to Reeb dynamics and celestial mechanics, including symmetry properties.

Abstract

In this article, we give two refinements of Franks' theorem: For orientation and area preserving homeomorphisms of the closed or open annulus, the existence of $k$-periodic orbits ($(k,n_0)=1$) forces the existence of infinitely many periodic orbits with periods prime to $n_0$. Moreover, if $f$ is reversible, the periodic orbits above could be symmetric. Our improvements of Franks' theorem can be applied to Reeb dynamics and celestial mechanics, for example, we give some precise information about the symmetries of periodic orbits found in Hofer, Wysocki and Zehnder's dichotomy theorem when the tight 3-sphere is equipped with some additional symmetries, and also the symmetries of periodic orbits on the energy level of H$\acute{e}$non-Heiles system in celestial mechanics.