Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra

Certain topological dynamical systems are considered that arise from actions of $\sigma$-compact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

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.

It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.'' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.

The translation action of ℝd on a translation bounded measure ω leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of ω, which is the carrier of the diffraction measure, lives on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) by van Enter and Miȩkisz (J. Stat. Phys. 66:1147–1153, 1992). Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of ‘atomic’ versus ‘molecular’ spectrum suggests a way to come to a more general correspondence between these two types of spectra.

Algorithm for determining pure pointedness of self-affine tilings

We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

The recently discovered Hat tiling [18] admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.

Pure point spectrum for measure dynamical systems on locally compact Abelian groups

The dynamical (or von Neumann) spectrum of a dynamical system and the diffraction spectrum of the corresponding measure dynamical system are intimately related. While their equivalence in the case of pure point spectra is well understood, this workshop aimed at an appropriate extension to systems with mixed spectra, building on recent developments for systems of finite local complexity and for certain random systems from the theory of point processes. Another focus was the question for connections between Schr¨odinger and dynamical spectra.

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.

Chapter 1. Role of statistics and data analysis. 1.1 Introduction. 1.2 Case studies. 1.3 Data. 1.4 Samples versus the population, some notation. 1.5 Vector and matrix notation. 1.6 Frequency distributions and histograms 1.7 The distribution as a model. 1.8 Sample moments. 1.9 Normal (Gaussian) distribution. 1.10 Exploratory data analysis. 1.11 Estimation. 1.12 Bias. 1.13 Causes of variance. 1.14 About data. 1.15 Reasons to conduct statistically based studies. 1.16 Data mining. 1.17 Modeling. 1.18 Transformations. 1.19 Statistical concepts. 1.20 Statistics paradigms. 1.21 Summary. 1.22 Exercises. Chapter 2. Modeling concepts. 2.1 Introduction. 2.2 Why construct a model? 2.3 What does a statistical model do? 2.4 Steps in modeling. 2.5 Is a model a unique solution to a problem? 2.6 Model assumptions. 2.7 Designed experiments. 2.8 Replication. 2.9 Summary. 2.10 Exercises. Chapter 3. Estimation and hypothesis testing on means and other statistics. 3.1 Introduction. 3.2 Independence of observations. 3.3 The Central Limit Theorem. 3.4 Sampling distributions. 3.4.1 t-distribution. 3.5 Confidence interval estimate on a mean. 3.6 Confidence interval on the difference between means. 3.7 Hypothesis testing on means. 3.8 Bayesian hypothesis testing. 3.9 Nonparametric hypothesis testing. 3.10 Bootstrap hypothesis testing on means. 3.11 Testing multiple means via analysis of variance. 3.12 Multiple comparisons of means. 3.13 Nonparametric ANOVA. 3.14 Paired data. 3.15 Kolmogorov-Smirnov goodness-of-fit test. 3.16 Comments on hypothesis testing. 3.17 Summary. 3.18 Exercises. Chapter 4. Regression. 4.1 Introduction. 4.2 Pittsburgh coal quality case study. 4.3 Correlation and covariance. 4.4 Simple linear regression. 4.5 Multiple regression. 4.6 Other regression procedures. 4.7 Nonlinear models. 4.8 Summary. 4.9 Exercises. Chapter 5. Time series. 5.1 Introduction. 5.2 Time Domain. 5.3 Frequency Domain. 5.4 Wavelets. 5.5 Summary. 5.6 Exercises. Chapter 6. Spatial statistics. 6.1 Introduction. 6.2 Data. 6.3 Three-dimensional data visualization. 6.4 Spatial association. 6.5 The effect of trend. 6.6 Semivariogram models. 6.7 Kriging. 6.8 Space-time models. 6.9 Summary. 6.10 Exercises. Chapter 7. Multivariate analysis. 7.1 Introduction. 7.2 Multivariate graphics. 7.3 Principal component analysis. 7.4 Factor analysis. 7.5 Cluster analysis. 7.6 Multidimensional scaling. 7.7 Discriminant analysis. 7.8 Tree based modeling. 7.9 Summary. 7.10 Exercises. Chapter 8. Discrete data analysis and point processes. 8.1 Introduction. 8.2 Discrete process and distributions. 8.3 Point processes. 8.4 Lattice data and models. 8.5 Proportions. 8.6 Contingency tables. 8.7 Generalized linear models. 8.8 Summary. 8.9 Exercises. Chapter 9 Design of experiments. 9.1 Introduction. 9.2 Sampling designs. 9.3 Design of experiments. 9.4 Comments on field studies and design. 9.5 Missing data. 9.6 Summary. 9.7 Exercises. Chapter 10 Directional data. 10.1 Introduction. 10.2 Circular data. 10.3 Spherical data. 10.4 Summary. 10.5 Exercises.

A class of improved estimators is proposed for N-point correlation functions of galaxy clustering and for discrete spatial random processes in general. In the limit of weak clustering, the variance of the unbiased estimator converges to the continuum value much faster than with any alternative, and all terms giving rise to a slower convergence exactly cancel. Explicit variance formulae are provided for both Poisson and multinomial point processes using techniques for spatial statistics reported by Ripley. The formalism naturally includes most previously used statistical tools such as N-point correlation functions and their Fourier counterparts, moments of counts in cells, and moment correlators. For all these, and perhaps some other statistics, our estimator provides a straightforward means for efficient edge corrections.

We consider (generalized) Kolakoski sequences on an alphabet with two even numbers. They can be related to a primitive substitution rule of constant length ℓ. Using this connection, we prove that they have pure point dynamical and pure point diffractive spectrum, where we make use of the strong interplay between these two concepts. Since these sequences can then be described as model sets with ℓ-adic internal space, we add an approach to “visualize” such internal spaces.