Closing Aubry sets I

Alessio Figalli (1) , Ludovic Rifford (2, 3)

1 Department of Mathematics and Statistics [Texas Tech]
2 McTAO - Mathematics for Control, Transport and Applications
3 LJAD - Laboratoire Jean Alexandre Dieudonné

Given a Tonelli Hamiltonian H : T∗M → R of class Ck, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x ̄, and a critical viscosity subsolution u, such that u is a C1 critical solution in an open neighborhood of the positive orbit of x ̄. Suppose further that u is "C2 at x ̄". Then there exists a Ck potential V : M → R, 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) If M is two dimensional, (1) holds replacing "C 1 critical solution + C 2 at x ̄" by "C 3 critical subsolution". These results can be considered as a first step through the attempt of proving the Man ̃ ́e's conjecture in C2 topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Man ̃ ́e's density Conjecture in C1 topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to prove C1-closing lemmas for dynamical systems.

Domaines: Systèmes dynamiques [math.DS]; Optimisation et contrôle [math.OC]

Alessio Figalli, Ludovic Rifford. Closing Aubry sets I. Communications on Pure and Applied Mathematics, 2015, 68 (2). ⟨hal-00935965⟩