Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

Abstract : We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F_n , G_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}.