Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments

Eduardo Garibaldi (1) , Samuel Petite (2) , Philippe Thieullen (3)

1 IMECC - Instituto de Matemática, Estatística e Computação Científica [Brésil]
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
3 IMB - Institut de Mathématiques de Bordeaux

The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we show the existence of these configurations for some environment of the dynamical system. Furthermore, in one dimension, we give sufficient conditions on the family of interaction energies to ensure, for any environment, the existence of calibrated configurations when the underlying dynamics is uniquely ergodic. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry-Mather theory and spaces of Delone sets

Mots clés: almost-periodic environment; Aubry-Mather theory; calibrated configuration; Delone set; Frenkel-Kontorova model; Mañé potential; Mather set; minimizing holonomic probability; weak KAM theory

Domaines: Systèmes dynamiques [math.DS]

Eduardo Garibaldi, Samuel Petite, Philippe Thieullen. Calibrated configurations for Frenkel-Kontorova type models in almost-periodic environments. 2013. ⟨hal-00915253v2⟩