  We study properties of action-minimizing invariant sets for Tonelli
Lagrangian and Hamiltonian systems and weak KAM solutions to the
Hamilton-Jacobi equation in terms of Mather's averaging functions. Our
principal discovery is that exposed points and extreme points of Mather's alpha
function are closely related to disjoint properties and graph properties of the
action-minimizing invariant sets, which is also related to $C^0$ integrability
of the systems and the existence of smooth weak KAM solutions to the
Hamilton-Jacobi equation.
