# Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals

## TL;DR

This paper proves the existence of equilibrium configurations in generalized Frenkel-Kontorova models on quasicrystals, extending classical models to aperiodic structures using almost-periodic potentials and anti-integrable limit techniques.

## Contribution

It introduces a framework for analyzing equilibrium states in Frenkel-Kontorova models on quasicrystals with almost-periodic potentials, broadening the scope beyond periodic cases.

## Key findings

- Existence of equilibrium configurations for prescribed rotation vectors.
- Applicability to classical and multidimensional Frenkel-Kontorova models.
- Use of anti-integrable limit method for proof.

## 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.