A particular minimization property implies C0-integrability

The manifold M being compact and connected and H being a Tonelli Hamiltonian such that T⁎M is equal to the dual tiered Mañé set, we prove that there is a partition of T⁎M into invariant C0-Lagrangian graphs. Moreover, among these graphs, those that are C1 cover a dense Gδ-subset of T⁎M. The dynamic restricted to each of these sets is non-wandering.