Closing Aubry sets II

Given a Tonelli Hamiltonian H : T ∗M → R of class C, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class C in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then, there exists a potential V : M → R of class Ck−1, small in C topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) For every > 0 there exists a potential V : M → R of class Ck−2, with ‖V ‖C1 < , for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. The latter result solves in the affirmative the Mañé density conjecture in C topology.