In this talk, we claim that the property of "non-degeneracy of all closed orbits of a given energy level" is a Mane generic property for non-convex Hamiltonians that are satisfying a certain geometric property.

A closed orbit is non-degenerate if its associated linearized Poincare map does not take roots of unity as an eigenvalue. Given a smooth Hamiltonian $H$ defined on cotangent bundle of a manifold, a property is called Mane generic if it holds for $H+u$, where $u$ is a generic potential. Perturbation of Hamiltonians in the sense of Mane is closely related to conformal perturbation of metrics.