A dichotomy theorem for minimizers of monotone recurrence relations

TL;DR

This paper establishes a dichotomy for minimizers of monotone recurrence relations, showing they are either well-behaved Birkhoff solutions or exhibit irregular, exponential growth under a strong twist condition.

Contribution

It proves a novel dichotomy theorem for global minimizers of monotone recurrence relations, extending Aubry-Mather theory to higher order and larger interaction ranges.

Findings

Minimizers are either Birkhoff or grow exponentially and oscillate.

Birkhoff minimizers satisfy strong ordering properties.

The dichotomy depends on a strong twist condition.

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 nontrivial 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 nonphysical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.