On the existence of infinitely many invariant Reeb orbits

In this article we extend results of Grove and Tanaka (Bull Am Math Soc 82:497-498, 1976, Acta Math 140:33-48, 1978) and Tanaka (J Differ Geom 17:171-184, 1982) 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 Lambda(M) asymptotically unbounded then for every fibrewise star-shaped hypersurface Sigma subset of T*M and every strict contactomorphism phi: Sigma -> Sigma which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.