Closing Aubry sets II

Abstract : Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class Ck+1 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 C2 topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) For every ǫ > 0 there exists a potential V : M → R of class Ck−2, with ∥V ∥C1 < ǫ, 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 Man ̃ ́e density conjecture in C1 topology.
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Alessio Figalli, Ludovic Rifford. Closing Aubry sets II. Communications on Pure and Applied Mathematics, Wiley, 2015, 68 (3). ⟨hal-00935970⟩

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