- 15-22 Xifeng Su, Lei Zhang, Rafael de la Llave
- Equilibrium quasi-periodic
configurations with resonant frequencies in quasi-periodic media I: perturbative expansions
Mar 11, 15
(auto. generated pdf),
of related papers
Abstract. 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
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.