A dynamical condition for differentiability of Mather's average action

Abstract

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.

Properties of action-minimizing sets and weak KAM solutions via Mather's averaging functions

“…Therefore, such disjoint properties of M c / A c / N c hold for infinitely many c ∈ H 1 (M ; R). We also remark that the condition (i) of Theorem 1.1 appears in the studies of asymptotically isolated invariant Lipschitz Lagrangian graph ( [11], Theorem 1.1) and of weak integrability ( [3], Theorem 1.1). We obtain the following corollary from Theorem 1.1.…”