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

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 M and is a non-trivial homotopy class of free loops in M. Then for almost every s ϵ [d,1) the level set {H = s} carries a periodic orbit z of the Hamiltonian system (T*M, ωᴏ,H) representing . Examples show that the condition {H < d} ᴐ  M is necessary and almost existence cannot be improved to everywhere existence.