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In discrete schemes, weak KAM solutions may be interpreted as approximationsof correctors for some Hamilton-Jacobi equations in the periodic setting. It isknown that correctors may not exist in the almost periodic setting. We show theexistence of discrete weak KAM solutions for non-degenerate and weakly twistinteractions in general. Furthermore, assuming equivariance with respect to alinearly repetitive quasi-periodic set, we completely classify all possibletypes of weak KAM solutions.

Classification of discrete weak KAM solutions on linearly repetitive quasi-periodic sets