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A dichotomy theorem for minimizers of monotone recurrence relations

Published online by Cambridge University Press:  27 September 2013

BLAŽ MRAMOR  and
BOB RINK
BLAŽ MRAMOR
Affiliation:
Institute of Mathematics, Albert-Ludwigs-Universität Freiburg, Germany email [email protected]
BOB RINK
Affiliation:
Department of Mathematics, VU University Amsterdam, The Netherlands email [email protected]
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Abstract

Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel–Kontorova model for a ferromagnetic crystal. For such problems, Aubry–Mather theory establishes the existence of ‘ground states’ or ‘global minimizers’ of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a non-trivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and non-physical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 1 , February 2015 , pp. 215 - 248
Copyright
© Cambridge University Press, 2013 

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