On a Theorem by Schlenk

Abstract

In this paper we prove a generalisation of Schlenk's theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk's theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. Either there exist two geometrically distinct noncontractible periodic Reeb orbits or the period of the noncontractible periodic Reeb orbit is small. This theorem can be applied to many physical systems including the H\'enon-Heiles Hamiltonian and Stark-Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on $\mathbb{R}^{2n}$ and that the Hofer metric is indeed a metric.