Minimal configurations for Frenkel-Kontorova model on a quasicrystal

TL;DR

This paper extends the analysis of the Frenkel-Kontorova model to quasicrystal substrates, proving the existence and continuity of rotation numbers for minimal configurations, generalizing classical crystalline results.

Contribution

It demonstrates that all non-negative real numbers can be rotation numbers of minimal configurations on quasicrystals, broadening previous crystalline substrate findings.

Findings

Every minimal configuration has a rotation number.

The rotation number varies continuously with configurations.

All non-negative real numbers are attainable as rotation numbers.

Abstract

In this paper, we consider the Frenkel-Kontorova model of a one dimensional chain of atoms submitted to a potential. This potential splits into an interaction potential and a potential induced by an underlying substrate which is a quasicrystal. Under standard hypotheses, we show that every minimal configuration has a rotation number, that the rotation number varies continuously with the minimal configuration, and that every non negative real number is the rotation number of a minimal configuration. This generalizes well known results obtained by S. Aubry and P.Y. le Daeron in the case of a crystalline substrate.