Rigidity in topology C⁰ of the Poisson bracket for Tonelli Hamiltonians

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 C⁰ converges on the compact subsets to a Tonelli Hamiltonian F. Let (G_n) be a sequence of Hamiltonians that that C⁰ converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F_n, G_n}) C⁰-converges on the compact subsets to a C¹ function $H$. Then $H=\{F,G\}$. (paper)