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

A dynamical condition for differentiability of Mather's average action