On the existence of infinitely many invariant Reeb orbits

TL;DR

This paper proves that on certain manifolds, there are infinitely many Reeb orbits invariant under specific contactomorphisms, extending prior results from geodesic to Reeb flow settings.

Contribution

It generalizes previous results on invariant geodesics to Reeb flows, establishing the existence of infinitely many invariant Reeb orbits under broad conditions.

Findings

Infinitely many invariant Reeb orbits exist under specified conditions.

Extension of invariant geodesic results to Reeb flows.

Applicable to manifolds with unbounded Betti numbers of free loop spaces.

Abstract

In this article we extend results of Grove and Tanaka on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space are asymptotically unbounded then for every fibrewise star-shaped hypersurface in the cotangent bundle of M and every strict contactomorphism of that hypersurface which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.