Existence of closed characteristics

Abstract

In the previous chapter, the dynamical approach to the symplectic invariants led to the special capacity function c0. Its construction was based on a variational principle for periodic solutions of certain Hamiltonian systems. The period of these periodic solutions was prescribed. In this chapter we shall deduce from this symplectic invariant the existence of periodic solutions on prescribed energy surfaces. If we neglect the parametrization of such solutions the aim is to find closed characteristics of a distinguished line bundle over a hypersurface in a symplectic manifold. It is determined by the symplectic structure, as explained in the introduction. A very special symplectic structure on the torus will lead us to M. Herman’s counterexample to the closing lemma in the smooth category.