Regular Model Sets Research Articles

Equivalence between pure point diffractive sets and cut-and-project sets on substitution tilings

Quasicrystals are characterized by the property of pure point diffractive spectrum mathematically. We look at substitution tilings and characterize the pure point diffractive spectrum by regular model sets defined from a cut-and-project scheme. The cut-and-project scheme is built with a physical space ℝ d and an internal space which is a product of a Euclidean space and a profinite group. The assumptions we make here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with same multiplicity. We give a precise argument for the proof on a specific example.

Amorphic complexity of group actions with applications to quasicrystals

In this article, we define amorphic complexity for actions of locally compact σ \sigma -compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for Z \mathbb {Z} -actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer’s cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.

Pure discrete spectrum and regular model sets on some non-unimodular substitution tilings.

Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut-and-project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600-610] is that unimodularity is no longer assumed in this paper.

Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.

Delone Sets Generated by Square Roots

Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note, we show that the point set given by the values with is a Delone set in the complex plane, for any . This complements Akiyama’s recent observation (see Spiral Delone sets and three distance theorem (2020), Nonlinearilty, 33(5): 2533–2540) that with forms a Delone set, if and only if α is badly approximated by rationals. A key difference is that our setting does not require Diophantine conditions on α.

Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings.

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

Diffraction of a model set with complex windows

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals.

Maximal equicontinuous generic factors and weak model sets

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice \begin{document}$ {\mathcal L}\subset G\times H $\end{document} and compact and aperiodic window \begin{document}$ W\subseteq H $\end{document} , have the maximal equicontinuous factor (MEF) \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} , if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [ 24 ]. For topological dynamical systems that possess a finite invariant measure with full support ( \begin{document}$ E $\end{document} -systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [ 33 ] and Auslander [ 2 ]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [ 34 ] that the centralizer of any \begin{document}$ {\mathcal B} $\end{document} -free dynamical system of Erdős type is trivial.

Diffraction of Return Time Measures

Letting $T$ denote an ergodic transformation of the unit interval and letting $f \colon [0,1)\to \mathbb{R}$ denote an observable, we construct the $f$-weighted return time measure $\mu_y$ for a reference point $y\in[0,1)$ as the weighted Dirac comb with support in $\mathbb{Z}$ and weights $f \circ T^z(y)$ at $z\in\mathbb{Z}$, and if $T$ is non-invertible, then we set the weights equal to zero for all $z < 0$. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of $\mu_{y}$ consists of a trivial atom and an absolutely continuous part, almost surely with respect to $y$. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations $T_{\alpha} \colon x \to x + \alpha \bmod{1}$ with rotation number $\alpha \in \mathbb{R}^+$. In contrast to when $T$ is mixing, we observe that the diffraction of $\mu_{y}$ is pure point, almost surely with respect to $y$. Moreover, if $\alpha$ is irrational and the observable $f$ is Riemann integrable, then the diffraction of $\mu_{y}$ is independent of $y$. Finally, for a converging sequence $(\alpha_{i})_{i \in \mathbb{N}}$ of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hof's classical definition beyond the class of amenable groups, and provide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

Pure Point Diffraction and Poisson Summation

We prove that the diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. This is achieved using Fourier analysis of unbounded measures on locally compact abelian groups as developed by Argabright and de Lamadrid. We also obtain diffraction results for certain classes of non-regular so-called weak model sets.

On weak model sets of extremal density

The theory of regular model sets is highly developed, but does not cover examples such as the visible lattice points, the kth power-free integers, or related systems. They belong to the class of weak model sets, where the window may have a boundary of positive measure, or even consists of boundary only. The latter phenomena are related to the topological entropy of the corresponding dynamical system and to various other unusual properties. Under a rather natural extremality assumption on the density of the weak model set, we establish its pure point diffraction nature. We derive an explicit formula that can be seen as the generalisation of the case of regular model sets. Furthermore, the corresponding natural patch frequency measure is shown to be ergodic. Since weak model sets of extremal density are generic for this measure, one obtains that the dynamical spectrum of the hull is pure point as well.

Characterization of model sets by dynamical systems

It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$ and then relate the properties of $\beta$ to ones of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.

Quasicrystals and model sets on substitution point sets

The typical property of a quasicrystal is that it lacks periodic order but its diffraction pattern has a strong component of Bragg peaks. We consider the diffraction pattern consisting purely of Bragg peaks and call it pure point diffraction. It has been known that any regular model sets are pure point diffractive. But the converse has been a conjecture. Here we show that in substitution point sets, the sets are pure point diffractive if and only if they are model sets.

Point Sets and Dynamical Systems In the Autocorrelation Topology

AbstractThis paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groupsG. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets ofGis complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced.The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to aG-dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as aG-dynamical system, is a factor of the dynamical local hull.

Dense Dirac combs in Euclidean space with pure point diffraction

Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact Abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean space, which is motivated by the study of quasicrystallographic random tilings.

Lattice substitution systems and model sets

This paper studies ways in which the sets of a partition of a lattice in ?n become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in ?n gives rise to regular model sets (based on p-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the n-dimensional chair tiling and the sphinx tiling are pure point diffractive.