Multidimensional constant-length substitution sequences

Frank, Natalie Priebe

doi:10.1016/j.topol.2004.08.014

Cited by 48 publications

(88 citation statements)

References 12 publications

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“…Note that D x = D y for x = y is possible. Since at most one prototile of a patch can have its left endpoint at a given position, it is clear that any D x can only have entries 0 and 1, namely These matrices are a generalisation of what is known as digit matrices in constant-length substitutions [56,28], wherefore we adopt the name here as well; compare also [47, Ch. VIII], where they appear as instruction matrices.…”

Section: Primitive Inflation Rules In One Dimensionmentioning

confidence: 99%

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Baake

Gähler

Mañibo

2019

Commun. Math. Phys.

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The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

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Section: Primitive Inflation Rules In One Dimensionmentioning

confidence: 99%

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Baake

Gähler

Mañibo

2019

Commun. Math. Phys.

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“…In Definition 1.2 of [Fra2], the author defines bijective substitutions. This definition reads in our notation as follows.…”

Section: Resultsmentioning

confidence: 99%

Computing Modular Coincidences for Substitution Tilings and Point Sets

Frettlöh

Sing

2007

Discrete Comput Geom

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Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in R d , consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the number of iterations needed. The main tool is a simple algorithm for computing modular coincidences, which is essentially a generalization of the Dekking coincidence to more than one dimension, and the proof of equivalence of this generalized Dekking coincidence and modular coincidence. As a consequence, we also obtain some conditions for the existence of modular coincidences. In a separate section, and throughout the article, a number of examples are given.

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“…Higher-dimensional examples with purely singular continuous spectrum, such as the squiral tiling [24] or similar bijective block substitutions [8], may still lead to classic Riesz products, though they are now in more than one variable, and the analysis is hence more involved. Nevertheless, the scaling analysis will still lead to a better understanding of such measures.…”

Section: Discussionmentioning

confidence: 99%

“…The singular continuous nature of γ is traditionally proved [4,8] by excluding pure points by Wiener's criterion [9,10] and absolutely continuous parts by the Riemann-Lebesgue lemma [11]; compare [2,6] and references therein for further material.…”

Section: Introductionmentioning

confidence: 99%