A dynamical condition for differentiability of Mather's average action

We prove the differentiability of $β$ 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 $β$ and local integrability of the Hamiltonian flow.

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