On the growth rate of Reeb orbits on fiberwise star-shaped hypersurfaces

In this talk, I will discuss the growth rate of Reeb orbits with respect to their period on certain contact manifolds. We focus on fiberwise star-shaped hypersurfaces in the cotangent bundle of a closed manifold. I will explain how a topological condition on the base manifold implies that the number of Reeb orbits with period at most T grows at least like the prime numbers, that is, on the order of T/logT. The topological condition is the existence of a certain non-nilpotent homology class of the free loop space (with respect to the Chas–Sullivan product). This is joint work with João Pering.