We consider Frenkel-Kontorova models corresponding to one-dimensional quasi-crystals. According to the arithmetic properties of the rotation numbers, we develop different arguments to show the existence of the associated quasi-periodic equilibria using Nash-Moser iterative method.
The results presented have an a-posteriori format. That is, 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. Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities, auxiliary equations and functional equations.