Closed Reeb orbits on the sphere and symplectically degenerate maxima

TL;DR

This paper proves that the existence of a special type of Reeb orbit on the sphere implies infinitely many periodic orbits, providing a new proof for a known theorem about Reeb flows on the three-sphere.

Contribution

It introduces a novel approach by adapting Hamiltonian Conley conjecture techniques to contact geometry, establishing a link between symplectically degenerate maxima and periodic orbits.

Findings

Existence of a symplectically degenerate maximum implies infinitely many Reeb orbits.

New proof of the theorem that Reeb flows on the standard three-sphere have at least two periodic orbits.

Methodology bridges Hamiltonian dynamics and contact geometry through adapted machinery.

Abstract

We show that the existence of one simple closed Reeb orbit of a particular type (a symplectically degenerate maximum) forces the Reeb flow to have infinitely many periodic orbits. We use this result to give a different proof of a recent theorem of Cristofaro-Gardiner and Hutchings asserting that every Reeb flow on the standard contact three-sphere has at least two periodic orbits. Our methods are based on adapting the machinery originally developed for proving the Hamiltonian Conley conjecture to the contact setting.