Periodic solutions on hypersurfaces and a result by C. Viterbo (Q1094720)

A parametrized family of compact hypersurfaces in \({\mathbb{R}}^{2N}\) modelled on a compact hypersurface S is a diffeomorphism \(\psi\) : ]- 1,1[\(\times S\to {\mathbb{R}}^{2N}\) onto an open bounded neighbourhood of S such that \(\psi (0,X)=X\) if \(X\in S\). The main result of the paper is the existence of a periodic orbit contained in \(\psi\) (]-\(\delta\),\(\delta\) [\(\times S)\) for every \(\delta >0\). Moreover, the corresponding action is bounded independently of \(\delta\). A direct consequence is Viterbo's theorem on the existence of a periodic orbit on every compact smooth hypersurface of contact type. The proof is related to Viterbo's proof, but simpler.