Anti-integrable limits for generalised Frenkel–Kontorova models on almost-periodic media

Abstract We study the equilibrium configurations for generalised Frenkel–Kontorova models subjected to almost-periodic media. By contrast with the spirit of the Kolmogorov–Arnold–Moser theory, our approach consists in establishing the other perturbation theory for fully chaotic systems far away from the integrable, which is called ‘anti-integrable’ limits. More precisely, we show that for large enough potentials, there exists a locally unique equilibrium with any prescribed rotation number/vector/plane, which is hyperbolic. The assumptions are general enough to satisfy both short-range and long-range Frenkel–Kontorova models and their multidimensional analogues.