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摘要
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Given a Tonelli Hamiltonian H:T*M of class C-k, with k2, 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 C-1 critical solution in an open neighborhood of the positive orbit of x. Suppose further that u is C-2 at x. Then there exists a C-k potential V : M, 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) If M is two dimensional, (1) holds replacing C-1 critical solution+C-2 at by CM3 critical subsolution. These results can be considered as a first step through the attempt of proving the Mane's conjecture in C-2-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 Mane's density conjecture in C-1-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 that were used to prove C-1-closing lemmas for dynamical systems.(c) 2014 Wiley Periodicals, Inc.
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CRITICAL SUB-SOLUTIONS
WEAK KAM THEOREM
TOTAL DISCONNECTEDNESS
MINIMIZING MEASURES
REGULARITY
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