  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.
