H. Koch, R. de la Llave, C. Radin Aubry-Mather theory for functions on lattices. (66K, Plain TeX) ABSTRACT. We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorovna models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.