Semi-discrete approximations converge in the Kuratowski sense to continuous Aubry and Mather sets, with full convergence under hyperbolicity for Aubry sets and finite ergodic measures for Mather sets.
Contreras, Action potential and weak KAM solutions,Calculus of Variations and Partial Differential Equations13(2001), no
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Semi-Discrete Approximation of Aubry and Mather sets
Semi-discrete approximations converge in the Kuratowski sense to continuous Aubry and Mather sets, with full convergence under hyperbolicity for Aubry sets and finite ergodic measures for Mather sets.