发表机构
University of Science and Technology of China; Queen Mary University of London; Anhui University of Science and Technology（中国科学技术大学; 伦敦大学玛丽女王学院; 安徽理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明字典序最大点为Sturmian序列的beta-移位不具有典型周期优化，并首次给出具有规范性质但不具有TPO的移位空间例子，主要利用Sturmian子移位的刚性性质。
AI 中文摘要
如果一个移位空间上的Lipschitz函数中，其唯一最大化测度支撑在周期轨道上的函数集合包含该Lipschitz函数空间的一个开稠密子集，则称该移位空间具有典型周期优化（TPO）。我们证明，字典序最大的点为Sturmian序列的beta-移位不具有TPO：在每个这样的beta-移位上，存在一个非空的Lipschitz函数开集，其中所有函数的唯一最大化测度均为Sturmian测度。这些是已知的第一类不具有TPO的beta-移位，并且由于它们具有规范性质，也是已知的第一类具有规范性质但不具有TPO的移位空间。相应的beta-变换在区间上的任何Hölder函数空间中都不具有TPO。证明这些结果的主要工具是Sturmian子移位的一个刚性性质：模去常数和Lipschitz余边界后，此类子移位上的Lipschitz函数空间是一维的。
英文摘要
A shift space is said to have typical periodic optimization (TPO) if the set of Lipschitz functions whose unique maximizing measure is supported on a periodic orbit contains an open dense subset of the space of Lipschitz functions. We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have TPO: on each such beta-shift there is a non-empty open set of Lipschitz functions, all of which have the Sturmian measure as their unique maximizing measure. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proof of these results is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional.
Comments33 pages
