# Closing Aubry sets II

created by figalli on 19 Jul 2013

[BibTeX]

Accepted Paper

Inserted: 19 jul 2013
Last Updated: 19 jul 2013

Journal: Comm. Pure Appl. Math.
Year: 2013

Abstract:

Given a Tonelli Hamiltonian $H:T^*M \rightarrow \R$ of class $C^k$, with $k\geq 4$, we prove the following results: (1) Assume there is a critical viscosity subsolution which is of class $C^{k+1}$ in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then, there exists a potential $V:M \rightarrow \R$ of class $C^{k-1}$, small in $C^2$ topology, for which the Aubry set of the new Hamiltonian $H+V$ is either an equilibrium point or a periodic orbit. (2) For every $\epsilon>0$ there exists a potential $V:M \rightarrow \R$ of class $C^{k-2}$, with $\ V\ _{C^1} < \epsilon$, 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 Ma\~n\'e density conjecture in $C^1$ topology.