Cut and project schemes, combinatorics and averaging properties for Toeplitz subshifts
Toeplitz 子移的剪切和投影方案、组合数学和平均属性
基本信息
- 批准号:454053022
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:WBP Fellowship
- 财政年份:2021
- 资助国家:德国
- 起止时间:2020-12-31 至 2022-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The project concerns various properties of subshifts associated to Toeplitz words. Here a "word" is a map from the integers into a finite set, that is, an infinite concatenation of symbols. In Toeplitz words, this concatenation is constructed in such a way that the word is not periodic, but still exhibits a certain degree of order. This makes them important examples in the theory of aperiodic order and interesting one-dimensional models for quasicrystals.In the context of quasicrystals, the spectrum of the associated Schrödinger operator plays an important role, since it encodes the energy levels that are available to electrons. If the so-called leading sequence condition (LSC) is satisfied, then the spectrum is a Cantor set of Lebesgue measure zero (actually it even implies uniformity of locally constant cocycles). However, the LSC has so far only been shown for subshifts of so-called simple Toeplitz words. One aim of the project is to extend this proof to a wider class of Toeplitz words. In addition it is planned to find sufficient or necessary conditions for general Toeplitz words to satisfy the LSC.Parts of the LSC are related to combinatorial properties of the involved words. Here an important step is to study cut and project schemes which can be associated to the Toeplitz word. They provide a way to generate the non-periodic, one-dimensional words from periodic, higher-dimensional lattices. Understanding how combinatorial properties of the word are related to properties of the projection set is therefore a first goal of the project; analysing the combinatorial properties themselves is a second one.The remaining part of the LSC deals with averages of matrix-valued functions along Toeplitz words, but the investigation of real-valued functions is planned as well. This topic is particularly interesting, since a recent preprint reduced the much studied Sarnak conjecture to averages over Toeplitz words with "high order" (entropy zero). The conjecture concerns averages of the so-called Möbius function, which play an important role in number theory.
该项目关注与Toeplitz词相关的子移位的各种属性。这里的“词”是从整数到有限集合的映射,也就是说,符号的无限连接。在Toeplitz的单词中,这种连接的构造方式使单词不是周期性的,但仍然显示出一定程度的顺序。这使它们成为非周期有序理论和有趣的准晶体一维模型中的重要例子。在准晶体的背景下,相关的Schrödinger算符的频谱起着重要的作用,因为它编码了电子可用的能级。如果满足所谓的导序列条件(LSC),则谱是Lebesgue测度为零的Cantor集(实际上它甚至意味着局部常共环的均匀性)。然而,到目前为止,LSC只显示了所谓的简单Toeplitz词的子移位。该项目的目的之一是将这一证明扩展到更广泛的Toeplitz词类。此外,还计划找出一般Toeplitz词满足LSC的充分或必要条件。LSC的部分内容与相关单词的组合属性有关。在这里,重要的一步是研究与Toeplitz单词相关的cut和project scheme。它们提供了一种从周期性高维晶格中生成非周期性一维词的方法。因此,了解单词的组合属性如何与投影集的属性相关是该项目的首要目标;分析组合特性本身是第二步。LSC的其余部分处理矩阵值函数沿Toeplitz词的平均值,但也计划研究实值函数。这个话题特别有趣,因为最近的一篇预印本将大量研究的Sarnak猜想简化为“高阶”(熵为零)的Toeplitz单词的平均值。该猜想涉及所谓Möbius函数的平均值,它在数论中起着重要作用。
项目成果
期刊论文数量(0)
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