The Weinstein conjecture with multiplicities on spherizations

TL;DR

This paper establishes lower bounds on the growth of closed Reeb orbits on fiberwise starshaped hypersurfaces in cotangent bundles, linking symplectic dynamics to the topology of the base manifold.

Contribution

It extends known results on geodesic growth to Reeb flows, considering different fundamental group growth scenarios, and generalizes theorems to broader symplectic contexts.

Findings

Lower bounds depend on the fundamental group's growth rate.

Generalization of geodesic growth theorems to Reeb flows.

Results apply to fiberwise starshaped hypersurfaces in cotangent bundles.

Abstract

Let M be a smooth closed manifold and T*M its cotangent bundle endowed with the usual symplectic structure. A hypersurface S in T*M is said to be fiberwise starshaped if for each point q in M the intersection of S with the fiber at q is starshaped with respect to the origin. In this thesis we give lower bounds of the growth rate of the number of closed Reeb orbits on a fiberwise starshaped hypersurface in terms of the topology of the free loop space of M. We distinguish the two cases that the fundamental group of the base space M has an exponential growth of conjugacy classes or not. If the base space M is simply connected we generalize the theorem of Ballmann and Ziller on the growth of closed geodesics to Reeb flows.