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We study properties of action-minimizing invariant sets for TonelliLagrangian and Hamiltonian systems and weak KAM solutions to theHamilton-Jacobi equation in terms of Mather's averaging functions. Ourprincipal discovery is that exposed points and extreme points of Mather's alphafunction are closely related to disjoint properties and graph properties of theaction-minimizing invariant sets, which is also related to $C^0$ integrabilityof the systems and the existence of smooth weak KAM solutions to theHamilton-Jacobi equation.

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