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.