KAM Theory For Quasi-Periodic Equilibria In 1-D Quasiperiodic Media And Aubry-Mather Theory For Generalized Frenkel-Kontorova Model

First, we consider Frenkel-Kontorova models corresponding to 1-D quasicrystals.We present a KAM theory for quasi-periodic equilibria. The theorem presented has an a-posteriori format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique.Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b) Bootstrap for regularity c) An efficient method to compute the breakdown of analyticity.Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms.Secondly, we consider Frenkel-Kontorova models in higher dimensional lattices. We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensionsWe also compare the proofs and results with those obtained in other formalisms.