Classification of Discrete Weak KAM Solutions on Linearly Repetitive Quasi-Periodic Sets

A discrete weak KAM solution is a potential function that highlights the ground state configurations at zero temperature of an infinite chain of atoms interacting with a periodic or quasi-periodic substrate. It is well known that weak KAM solutions exist for periodic substrates as in the Frenkel–Kontorova model. Weak solutions may not exist in the almost periodic setting as in the theory of stationary ergodic Hamilton–Jacobi equations (where they are called correctors). For linearly repetitive quasi-periodic substrates, we show that equivariant interactions that fulfill a twist condition and a non-degenerate property always admit sublinear weak KAM solutions. We moreover classify all possible types of weak KAM solutions and calibrated configurations according to an intrinsic prefered order. The notion of prefered order is new even in the classical periodic case.