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January  1997, 3(1): 135-151. doi: 10.3934/dcds.1997.3.135

Aubry-Mather theory for functions on lattices

Hans Koch 1, , Rafael De La Llave 2, and  Charles Radin 1,

1. 

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, United States
2. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States

Received  October 1996 Published  October 1996

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: functions on lattices., Aubry-Mather theory.
Mathematics Subject Classification: 58F99, 82B20, 49J4.
Citation: Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
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