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.
Barles,Solutions de viscosité des équations de Hamilton–Jacobi, Mathematics and Its Applications, vol
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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.