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Wigner measures supported on weak KAM tori
Authors:Alberto Parmeggiani  Lorenzo Zanelli
Institution:1. Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126, Bologna, Italy
2. Department of Mathematics, University of Padova, Via Trieste 63, 35121, Padova, Italy
Abstract:In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \) , we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures.
Keywords:
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