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Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments

Abstract

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 focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. 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.

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Correspondence to Samuel Petite.

Additional information

Work is supported by FAPESP 2009/17075-8, Brazilian-French Network in Mathematics CAPES-COFECUB 661/10, MAth AmSud 38889TM—DCS and ANR WKBHJ “Weak KAM” ANR-12-BS01-0020.

Communicated by Dmitry Dolgopyat.

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Garibaldi, E., Petite, S. & Thieullen, P. Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments. Ann. Henri Poincaré 18, 2905–2943 (2017). https://doi.org/10.1007/s00023-017-0589-7

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