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 good models for quasicrytals, 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 analogues. The proof uses the idea of the anti-integrable limit.