Action and index spectra and periodic orbits in Hamiltonian dynamics

Abstract

The paper focuses on the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits.

We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action-index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of ${ℂ\mathbb{P}}^n$ with exactly $n+1$ periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on ${ℂ\mathbb{P}}^n$ is normalized to be in the same cohomology class as the first Chern class.