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. In this case, the lower bound for the displacement energy is explicitly given by the difference of the two periods. This theorem can be applied to many physical systems including the Hénon–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 R2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{2n}$$\\end{document} and that the Hofer metric is indeed a metric.