Minimizing Orbits in the Discrete Aubry-Mather Model

We consider a generalization of the Frenkel-Kontorova model in higher dimension. We give a wider applicability to Aubry’s theory by studying models with vector-valued states over a one dimensional chain. This theory has a lot of similarities with Mather’s twist approach over a multidimensional torus. Weakening the standard hypotheses used in one dimensional, we investigate properties (like boundness of jumps and definability of a rotation vector) of a special class of strong ground states: the calibrated configurations.The main mathematical tool is to cast the study the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions in the same spirit of Hamilton-Jacobi methods. With reduced hypotheses, we reproduce in this discrete setting some classical results of the Lagrangian Aubry-Mather theory. In particular, we obtain a graph property for the Aubry set, representation formulas for calibrated sub-actions and the existence of separating sub-actions.