Minimal Configurations in the Aubry-Mather Theory

  • Simeon Reich
  • Alexander J. Zaslavski
Part of the Developments in Mathematics book series (DEVM, volume 34)


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.


Energy Function Rational Number Dense Subset Nonnegative Function Rotation Number 
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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Simeon Reich
    • 1
  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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