Genericity in Nonlinear Analysis pp 481-512 | Cite as

# Minimal Configurations in the Aubry-Mather Theory

## Abstract

Chapter 10 is devoted to the Aubry-Mather theory applied to the famous Frenkel-Kontorova model, an infinite discrete model of solid-state physics related to dislocations in one-dimensional crystals. In this model a configuration of a system is a sequence of real numbers with indices from −∞ to +∞. We are interested in (*h*)-minimal configurations with respect to an energy function *h*. A configuration is called (*h*)-minimal if its total energy cannot be made less by changing its final states. Classical Aubry-Mather theory is concerned with finding and investigating *h*-minimal configurations with a given rotation number, where the function *h* is fixed. It implies that the set of all periodic *h*-minimal configurations of a rational rotation number *p*/*q* is totally ordered. Moreover, between any two neighboring periodic *h*-minimal configurations with rotation number *p*/*q*, there are (non-periodic) *h*-minimal heteroclinic connections having the same rotation number *p*/*q*. We consider a complete metric space of energy functions *h* equipped with a certain *C* ^{2} topology and show that for most energy functions in this space, there exist three different *h*-minimal configurations with rotation number *p*/*q* such that any other *h*-minimal configuration with the same rotation number *p*/*q* is a translation of one of these three.

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