Minimal configurations for Frenkel-Kontorova model on a quasicrystal

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.

25 pages, 8 figures

References in corpus (3)

• Tiling Spaces are Inverse Limits
• On the Dynamics of G-Solenoids. Applications to Delone Sets
• Spaces of tilings, finite telescopic approximations and gap-labelling

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• Resonant Equilibrium configurations in quasi-periodic media: perturbative expansions
• KAM theory for quasi-periodic equilibria in 1-D quasiperiodic media
• Critical states of strongly interacting many-particle systems on a circle
• On the existence of solutions for Frenkel-Kontorova models on quasi-crystals
• Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals