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

2020 ◽  
Vol 76 (5) ◽  
pp. 600-610
Author(s):  
Dong-il Lee ◽  
Shigeki Akiyama ◽  
Jeong-Yup Lee
Keyword(s):  
Discrete Spectrum ◽  
Additional Condition ◽  
Internal Space ◽  
Representative Point ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Regular Model ◽  
Model Sets ◽  
Project Scheme ◽  
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.

Point Sets and Dynamical Systems In the Autocorrelation Topology

2004 ◽  
Vol 47 (1) ◽  
pp. 82-99 ◽  
Author(s):  
Robert V. Moody ◽  
Nicolae Strungaru
Keyword(s):  
Dynamical System ◽  
Point Sets ◽  
Finite Point ◽  
Locally Finite ◽  
Uniform Topology ◽  
Regular Model ◽  
Model Sets ◽  
Project Scheme ◽  
Compact Abelian Groups ◽  
Delone Sets

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.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 511 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

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.

scholarly journals Cut-and-Project Schemes for Pisot Family Substitution Tilings

2018 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Shigeki Akiyama ◽  
Yasushi Nagai
Keyword(s):  
Point Spectrum ◽  
The Other ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Internal Space ◽  
Sufficient Condition ◽  
Abstract Space ◽  
Dynamical Spectrum ◽  
Substitution Tiling ◽  
Substitution Tilings

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].

On the frequency module of the hull of a primitive substitution tiling

2022 ◽  
Vol 78 (1) ◽  
Author(s):  
April Lynne D. Say-awen ◽  
Dirk Frettlöh ◽  
Ma. Louise Antonette N. De Las Peñas
Keyword(s):  
Statistical Properties ◽  
Absolute Frequency ◽  
Aperiodic Tilings ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Tiling Spaces ◽  
The Family ◽  
Frequency Module ◽  
Substitution Tilings ◽  
The Absolute

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

FRACTAL TILINGS FROM SUBSTITUTION TILINGS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950009
Author(s):  
XINCHANG WANG ◽  
PEICHANG OUYANG ◽  
KWOKWAI CHUNG ◽  
XIAOGEN ZHAN ◽  
HUA YI ◽  
...  
Keyword(s):  
Short Paper ◽  
Self Similarity ◽  
Substitution Rule ◽  
Substitution Tiling ◽  
Simple Strategy ◽  
Substitution Tilings

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

scholarly journals Three variations on a theme by Fibonacci

2020 ◽  
pp. 2140001
Author(s):  
Michael Baake ◽  
Natalie Priebe Frank ◽  
Uwe Grimm
Keyword(s):  
Direct Product ◽  
Two Dimensions ◽  
Pure Point ◽  
One Dimension ◽  
Stochastic Nature ◽  
Regular Model ◽  
Model Sets ◽  
Planar Extension ◽  
Dynamical Spectra ◽  
Variations On A Theme

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.

Uniform Distribution in Model Sets

2002 ◽  
Vol 45 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Uniform Distribution ◽  
Physical Space ◽  
Pure Point ◽  
Internal Space ◽  
Compact Closure ◽  
Related Model ◽  
Model Sets ◽  
Distribution Property ◽  
Distribution Of Points ◽  
Model Set

AbstractWe give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining themodel set ismeasurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

scholarly journals On substitution tilings of the plane with n-fold rotational symmetry

2015 ◽  
Vol Vol. 17 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Gregory R. Maloney
Keyword(s):  
Rotational Symmetry ◽  
Substitution Tiling ◽  
Computer Assistance ◽  
Discrete Algorithms ◽  
Special Cases ◽  
International Audience ◽  
Substitution Tilings

Discrete Algorithms International audience A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and includes various special cases that have already been constructed by hand for low values of n. An example constructed by this method for n = 11 is exhibited; this is the first substitution tiling with elevenfold symmetry appearing in the literature.

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