  We consider 1-D quasi-periodic Frenkel-Kontorova models.
  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 equilibria are orbits of a dynamical system
which has very unusual properties. We obtain results on the Lyapunov exponents
of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic
equilibria. We show that the equilibria can be pinned even if the gap is zero.
