单调递归关系极小元的一个二分法
A dichotomy theorem for minimizers of monotone recurrence relations
摘要：
变分单调递归关系出现在固态物理中，作为铁磁晶体的Frenkel-Kontorova模型的推广。对于这样的问题，Aubry-Mather理论建立了任意旋转数的“基态”或“全局极小值”的存在。最近邻晶体模型等价于哈密顿扭曲映射。在这种情况下，全局极小点有一个特殊的性质：它们只能交叉一次。作为一个重要的结果，它们中的每一个都具有伯克霍夫性质。在具有较大范围的相互作用和高阶递归关系的晶体中，单交叉性质不成立，并且可以存在不是Birkhoff的全局极小值。本文研究了全局极小点的交叉问题。在强扭曲条件下，我们证明了以下二分法：它们要么是Birkhoff，因此非常规则，要么是非常不规则和非物理的：然后它们以指数方式增长并振荡。对于Birkhoff极小化，我们还证明了一些强有序性是众所周知的扭曲映射。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
