Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

Abstract

I study classes of generalized Frenkel–Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as models for quasicrystals, this setup presents Frenkel–Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman’s discovery of quasicrystals. Here I consider models with configurations $u:{\mathbb {Z}}^r \rightarrow {\mathbb {R}}^d$, where d is the dimension of the quasicrystal, for any r and d. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $C^2$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel–Kontorova models and its multidimensional analoges. The proof uses the idea of the anti-integrable limit.