Geometric Realizations of Two-Dimensional Substitutive Tilings

Bédaride, Nicolas; Hilion, Arnaud

doi:10.1093/qmath/has025

Cited by 9 publications

(15 citation statements)

References 9 publications

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“…, b d }, the equation (5) in Theorem 4.12 holds. The equation (5) in Theorem 4.12 also holds for a return vector z and x ∈ {(φ * ) −kn b i | k ∈ Z >0 , i = 1, 2, . .…”

Section: Necessary Results For Self-affine Tilingsmentioning

confidence: 95%

Distribution of patches in tilings and spectral properties of corresponding dynamical systems

Nagai

2015

Nonlinearity

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A tiling is a cover of R d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our results throw light on inverse problem of deducing information of tilings from information of diffraction measures, in a quite general setting.

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“…, b d }, the equation (5) in Theorem 4.12 holds. The equation (5) in Theorem 4.12 also holds for a return vector z and x ∈ {(φ * ) −kn b i | k ∈ Z >0 , i = 1, 2, . .…”

Section: Necessary Results For Self-affine Tilingsmentioning

confidence: 95%

Distribution of patches in tilings and spectral properties of corresponding dynamical systems

Nagai

2015

Nonlinearity

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“…We remark that the study of such non-periodic tilings in the hyperbolic plane has a long history (see e.g. [5,11,24,26]), but we will see that hyperbolic model sets and their associated tilings have a number of features which are not known to hold in previous examples.…”

Section: Tilings Of the Hyperbolic Planementioning

confidence: 85%

“…Definition 6. 5 The pair (G, K ) is called a Gelfand pair and K \G is called a commutative space if the Hecke algebra C c (G, K ) is commutative under convolution.…”

Section: Approximation Of the Auto-correlation For Weighted Regular Model Setsmentioning

confidence: 99%

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

Björklund

Hartnick²,

Pogorzelski

2021

Math. Z.

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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 symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $$\mathbb {R}^n$$ R n . This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.

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“…Remark 2.8. Usually tiles are defined to be (1) a compact set that is the closure of its interior [5], or in Euclidean case, (2) a polygonal subset of R d [17] or (3) a homeomorphic image of closed unit ball (for example, [1]). The advantage of our definition is that we can give punctures to tiles and we do not need to consider labels (Example 2.9), and so we may avoid a slight abuse of language such as "tiles T and S have disjoint interiors" and simplify the notation.…”

Section: Proofmentioning

confidence: 99%

A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation

Nagai

2019

Discrete Comput Geom

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We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in the literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object P to one that is MLD with P. Moreover we show that, for an object P and a class Σ of objects, a mild condition on them assures that there exists some Q ∈ Σ that is MLD with P. As an application, we study pattern equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information of the original object up to MLD in a quite general setting.By this operation we forget the behavior of T outside C. 2. Some of the objects "include" other objects. For patches this means the usual inclusion of two sets; for measures this means one measure is a restriction of another. 3. They admit glueing operation. For example, suppose {P i | i ∈ I} is a family of patch such that if i, j ∈ I, T ∈ P i and S ∈ P j , then either S = T or S ∩ T = ∅. Then we can "glue" P i 's and obtain a patch i∈I P i .

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Geometric Realizations of Two-Dimensional Substitutive Tilings

Cited by 9 publications

References 9 publications

Distribution of patches in tilings and spectral properties of corresponding dynamical systems

Distribution of patches in tilings and spectral properties of corresponding dynamical systems

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

A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation

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