纯点衍射置换Delone集具有Meyer性质

A Note on Entropy of Delone Sets

In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non-compact) locally compact Abelian groups that contain Meyer sets.

自相似德洛内集与皮索特数

Self-similar Delone sets and Pisot numbers

本文研究自相似德洛内集，证明温和条件下相似因子为皮索特数等价于模式一致离散，还分析了迈耶集相关情形及给出示例，为相关模式研究提供了理论依据。

对于具有均匀簇频率的Delone集，纯点衍射隐含零熵

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

研究了欧几里得空间中具有有限局部复杂度的Delone集.我们表明，这样的一个集有补丁计数和拓扑熵0，如果它有统一的集群频率和纯点衍射。我们还注意到，当重复率函数满足一定的增长限制时，斑块计数熵为0。

On substitution tilings and Delone sets without finite local complexity

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion…

Meyer集、拓扑特征值和Cantor纤维丛

Meyer sets, topological eigenvalues, and Cantor fiber bundles

我们引入了Meyer集的两个新的刻画。局部复杂度有限的$\R^d$中的重复Delone集与Meyer集拓扑共轭当且仅当它具有$d$线性无关的拓扑特征值，即当且仅当它与具有全不连通的紧纤维和扩张正则作用的$d$ -环面上的丛拓扑共轭.“共轭”是一个非平凡的条件，因为我们表明，存在集的拓扑共轭迈耶集，但本身不是迈耶。我们还展示了一个衍射集，是不是迈耶，回答在否定的一个问题所提出的Lagarias，并展示了迈耶集的可测量和拓扑特征值是不同的。

Delone集的周期性与局部复杂性

An Algebraic Approach to Periodicity and Complexity of Delone Sets

本文通过代数方法研究Delone集的复杂性与周期性，推广了零化子与周期化子概念，并证明了低复杂度配置及Meyer集存在非平凡零化子，同时给出了强制周期性的条件。