An almost existence theorem for non-contractible periodic orbits in cotangent bundles

TL;DR

This paper proves that under certain conditions, almost every energy level in a cotangent bundle contains a non-contractible periodic orbit representing a given homotopy class, extending the understanding of Hamiltonian dynamics.

Contribution

It establishes an almost existence theorem for non-contractible periodic orbits in cotangent bundles, highlighting the necessity of specific geometric conditions.

Findings

Almost every energy level above a threshold contains a non-contractible periodic orbit.

The condition that the sublevel set contains the zero section is necessary.

Almost existence cannot be extended to all energy levels without additional assumptions.

Abstract

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H<d} contains the zero section and \alpha is a non-trivial homotopy class of free loops in M. Then for almost every s>=d the level set {H=s} carries a periodic orbit z of the Hamiltonian system (T^*M,\omega_0,H) representing \alpha. Examples show that the condition that {H<d} contains M is necessary and almost existence cannot be improved to everywhere existence.