A dynamical condition for differentiability of Mather's average action

We prove the differentiability of $\beta $ of Mather function on all homology classes corresponding to rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian absorbing graph, invariant by Tonelli Hamiltonians. We also show the relationship between local differentiability of $\beta $ and local integrability of the Hamiltonian flow.