On the existence of infinitely many invariant Reeb orbits
  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. 
Will J. Merry, Kathrin Naef
math.SG