Aubry-Mather theory for functions on lattices

Hans Koch; Rafael De La Llave; Charles Radin

doi:10.3934/dcds.1997.3.135

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1997, Volume 3, Issue 1: 135-151. Doi: 10.3934/dcds.1997.3.135

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Aubry-Mather theory for functions on lattices

1.
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712
2.
Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712

Received: September 30, 1996

Published: December 1996

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Abstract

We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.

Keywords:

Aubry-Mather theory,
functions on lattices.

Mathematics Subject Classification: 58F99, 82B20, 49J40.

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