Spiral Delone sets in relative metric

Number of bounded distance equivalence classes in hulls of repetitive Delone sets

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].

Delone sets and dynamical systems

In these expository notes we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.

Equicontinuous factors, proximality and Ellis semigroup for Delone sets

We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.

A measure theoretic result for approximation by Delone sets

With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish a full converse of the Borel-Cantelli lemma. This provides an analogue of more classical problems in the metric theory of Diophantine approximation, but with the distance to the nearest integer function replaced by distance to an arbitrary Delone set.

Chaotic Delone sets

We present a definition of chaotic Delone set, and establish the genericity of chaos in the space of $(ε,δ)$-Delone sets for $ε\geq δ$. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.

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.

Delone集的双Lipschitz与有界位移等价性

BiLipschitz and bounded displacement equivalence of Delone sets

本文综述Delone集的双Lipschitz与有界位移等价性，重点研究非周期序相关实例，涉及经典结果、相关判据及替换铺砌、截集等类别的扩展情况。

Pure point diffractive substitution Delone sets have the Meyer property

We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J. C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra.

Some examples of non-rectifiable, repetitive Delone sets

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non bi-Lipschitz equivalent to the standard lattice). More generally, we construct such a set so that the translation action on the closure of its orbit has any prescribed Choquet simplex as its set of invariant probability measures (in particular, we provide uniquely ergodic examples). The construction relies on classical examples of Burago-Kleiner and McMullen, …

Diffusion on Delone sets

We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.

Linearly repetitive Delone sets are rectifiable

In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $λ\ZZ^d$ for some positive $λ$. This condition includes primitive Pisot substitution tilings but also concern…

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

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.

Tower systems for Linearly repetitive Delone sets

In this paper we study linearly repetitive Delone sets and prove, following the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly repetitive Delone set admits a properly nested sequence of box decompositions (tower system) with strictly positive and uniformly bounded (in size and norm) transition matrices. This generalizes a result of Durand for linearly recurrent symbolic systems. Furthermore, we apply this result to give a new proof of a classic estimation of Lagarias and Pleasants on the rate of convergence of patch-frequencies.

自仿射Delone集与偏差现象

Self affine Delone sets and deviation phenomena

本文研究自仿射Delone集上平移作用的遍历积分增长，证明其增长率由重整化映射在模式空间上同调上的诱导作用控制，并探讨了对衍射的影响及相关例子。

A new relation between geometric and dynamical properties of objects such as tilings and Delone sets

Let P be an object such as tiling, Delone set and weighted Dirac comb. There corresponds a dynamical system to P, called the corresponding dynamical system. Such dynamical systems are geometric analogues of symbolic dynamics. It is well-known that there are correspondences between geometric properties of P and properties of the corresponding dynamical system. In this article we give a new correspondence. In other words, we characterize the property that the group of topological eigenvalues for the corresponding dynamical system is not discrete, in terms of a geometric property of P.

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…

Canonical diffusions on the pattern spaces of aperiodic Delone sets

We consider pattern spaces of aperiodic and repetitive Delone sets of finite local complexity. These spaces are compact metric spaces and constitute a special class of foliated spaces. We define new Sobolev spaces with respect to the unique invariant measure and prove the existence of the unitary Schrödinger semigroup, which in physics terms describe the evolution of phasons. We define and study natural leafwise diffusion processes on these pattern spaces. These processes have Feller, but lack strong Feller and hypercontractivity properties, and heat kernels do not exist. The associated Dirich…

Dynamical properties of almost repetitive Delone sets

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense…

On the Dynamics of G-Solenoids. Applications to Delone Sets

A G-solenoid is a laminated space whose leaves are copies of a single Lie group G, and whose transversals are totally disconnected sets. It inherits a G-action and can be considered as dynamical system. Free Z^d-actions on the Cantor set as well as a large class of tiling spaces possess such a structure of G-solenoid. We show that a G-solenoid can be seen as a projective limit of branched manifolds modeled on G. This allows us to give a topological description of the transverse invariant measures associated with a G-solenoid in terms of a positive cone in the projective limit of the dim(G)-hom…

与自仿射 Delone 集相关的随机算子的迹与 Shubin 公式

Traces of random operators associated with self-affine Delone sets and Shubin's formula

本文研究自仿射 Delone 集上算子的迹，证明有限维限制的迹受渐近循环定义的迹控制，并据此改进 Shubin 公式、刻画其收敛误差。

Delone dynamical systems and spectral convergence

In the realm of Delone sets in locally compact, second countable, Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty-Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topologic…

Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. T…

在庞加莱盘上研究切和投影方案：从紧致富士扬群到混沌德洛内集

Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

本文研究了基于紧致富士扬群的切和投影方案，证明了其生成的德洛内集为混沌结构，并扩展了相关研究。

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。

Deformation of Delone dynamical systems and pure point diffraction

This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under ``equivariant'' local perturbations and discuss various examples,including deformed model sets. A key step in the proof of stability consists in transforming the problem into …

阿贝尔群之外的线性重复性

Linear repetitivity beyond abelian groups

本文将线性重复 Delone 集包络的唯一遍历性推广到阿贝尔群之外，并借助 Ornstein–Weiss 拟铺砌与次可加收敛定理，在可和幺模 lcsc 群中建立缓增重复条件下的相应结论。

欧几里得空间之外的纯点衍射与熵

Pure point diffraction and entropy beyond the Euclidean space

本文将纯点衍射 Delone 集的零熵性质推广到 σ-紧局部紧 Abel 群，证明相关动力系统拓扑熵为零，并构造反例说明补片计数熵可能非零、不可沿极限计算甚至无限。

Aperiodic order and spherical diffraction, II: Translation bounded measures on homogeneous spaces

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian …

Toeplitz flows and model sets

We show that binary Toeplitz flows can be interpreted as Delone dynamical systems induced by model sets and analyse the quantitative relations between the respective system parameters. This has a number of immediate consequences for the theory of model sets. In particular, we use our results in combination with special examples of irregular Toeplitz flows from the literature to demonstrate that irregular proper model sets may be uniquely ergodic and do not need to have positive entropy. This answers questions by Schlottmann and Moody.

斯特鲁米安晶格与非周期瓷砖集

Sturmian lattices and Aperiodic tile sets

本文提出基于二次斜率斯特鲁米安词构造非周期性瓷砖集的算法，并利用斯特鲁米安晶格和Delone集的有界位移等价性进行分类和构造。

Pure Point Dynamical and Diffraction Spectra

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

Conjugacies of model sets

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity (FLC) that is topologically conjugate to $M$ is mutually locally derivable (MLD) to a model set $M'$ that has the same internal group and window as $M$, but has a different projection from $H \times R^d$ to $R^d$. In cohomological terms, this means that the group $H^1_{an}(M,R)$ of asymptotically negligible classes has dimension $n$. We also exhibit a counter…

Meyer sets, topological eigenvalues, and Cantor fiber bundles

We introduce two new characterizations of Meyer sets. A repetitive Delone set in $\R^d$ with finite local complexity is topologically conjugate to a Meyer set if and only if it has $d$ linearly independent topological eigenvalues, which is if and only if it is topologically conjugate to a bundle over a $d$-torus with totally disconnected compact fiber and expansive canonical action. "Conjugate to" is a non-trivial condition, as we show that there exist sets that are topologically conjugate to Meyer sets but are not themselves Meyer. We also exhibit a diffractive set that is not Meyer, answerin…

来自斯图姆晶格的非周期平铺集

Aperiodic tile sets from Sturmian lattices

研究基于二次斜率的斯图姆单词构造非周期平铺集，核心方法是给出显式算法，关键要素有斯图姆晶格和德洛内集的有界位移等价性，能生成基础缩放常数为实二次域单位的非周期平铺集。

一条线上的均匀分布Kakutani铺砌异常集合

An exceptional set of uniformly spread Kakutani tilings of the line

研究发现五个特定参数下，Kakutani铺砌能形成均匀分布的晶格位移，结合多项式分类与替代规则分析得出这一结论。

完全Gabor框架、其存在性问题以及一个非均匀Balian-Low型定理

Full Gabor frames, its existence problem, and a non-uniform Balian-Low type theorem

针对一类在数学和物理中重要的Delone集，证明了非均匀Balian-Low型定理并解决了Gabor框架存在的逆问题，引入完全Gabor框架并证明其存在等价于下Beurling密度严格大于1。