On the existence of infinitely many invariant Reeb orbits

Abstract

Abstract. In this article we extend results of Grove and Tanaka [GT76,GT78,Tan82] 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 Λ(M ) are asymptotically unbounded then for every fibrewise star-shaped hypersurface Σ ⊂ T * M and every strict contactomorphism ϕ : Σ → Σ which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.