Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics

TL;DR

This paper explores the relationship between periodic orbits and the action/index spectrum in Hamiltonian systems, proving the existence of infinitely many orbits under certain conditions and analyzing their spectral properties.

Contribution

It establishes new results linking action and index spectra to the existence and bounds of periodic orbits in Hamiltonian dynamics on rational manifolds.

Findings

Infinitely many periodic orbits exist on certain rational manifolds.

The minimal action--index gap remains bounded in specific cases.

For projective spaces with n+1 orbits, action and mean index differences are constant.

Abstract

The main theme of this paper is 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 the n-dimensional complex projective space 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 the projective space is normalized to be in the same cohomology class as the first Chern class.