A dichotomy theorem for minimizers of monotone recurrence relations

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

Positive topological entropy for monotone recurrence relations

“…We remark that the twist condition (H4) is weaker than that in [20] in that we require ∂ i,k h ≤ 0 while in [20] it is assumed that ∂ i,k h ≡ 0, for i = 1, k = 1, and i = k. So we cannot simply apply directly the conclusions of [20], especially the dichotomy theorem for minimizers. Nevertheless, inspired by the ideas of Mramor and Rink in [20,21] and Bangert in [5,6] (we even borrow most of the notation from [20]) we obtain the conclusion we need that a minimizer with bounded action is Birkhoff.…”

“…Nevertheless, inspired by the ideas of Mramor and Rink in [20,21] and Bangert in [5,6] (we even borrow most of the notation from [20]) we obtain the conclusion we need that a minimizer with bounded action is Birkhoff. …”

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

“…Birkhoff minimizers or nonself-intersecting minimal solutions with any rotation vector have been constructed. However, as observed by Blank [1989], there exist minimizers that are not Birkhoff for high-dimensional lattice systems, see [Mramor & Rink, 2013] for such an example for the generalized FK model with non-nearest neighbor interactions. Similarly, for variational problems on a torus, there are minimal solutions with self-intersections, as remarked by Moser [1986].…”

“…Therefore, one needs new approaches to study the properties of minimizers or Birkhoff solutions [Mramor & Rink, 2012, 2013Wang & Qin, 2014a, 2014b. It was shown by Mramor and Rink [2013] that under some conditions, each minimizer is either Birkhoff, or it is oscillating and exponentially growing. We proved in [Guo et al, 2014] that this conclusion holds true even if we relax the strong twist condition in [Mramor & Rink, 2013].…”

“…It was shown by Mramor and Rink [2013] that under some conditions, each minimizer is either Birkhoff, or it is oscillating and exponentially growing. We proved in [Guo et al, 2014] that this conclusion holds true even if we relax the strong twist condition in [Mramor & Rink, 2013]. Precisely, we showed that for the one-dimensional generalized FK model each minimizer with bounded action is Birkhoff.…”