Equilibrium Quasi-periodic Configurations with Resonant Frequencies in Quasi-periodic Media I: Perturbative Expansions

We consider 1-D quasi-periodic Frenkel-Kontorova models (describing, for example, deposition of materials in a quasi-periodic substratum). We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that the dynamical system has very unusual properties. Using these, we obtain results on the Lyapunov exponents of the resonant quasi-periodic solutions. In a companion paper, we develop a rather unusual KAM theory (requiring new considerations) which establishes that the perturbative expansions converge when the perturbing potentials satisfy a one-dimensional constraint. Quasi-periodic Frenkel-Kontorova models, resonant frequencies, equilibria, quasicrystals, Lindstedt series, counterterms [2010] 70K43, 37J50, 37J40, 52C23