Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

Arnoux, Pierre; Berthé, Valérie; Ei, Hiromi; Ito, Shunji

doi:10.46298/dmtcs.2291

Cited by 23 publications

(10 citation statements)

References 21 publications

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“…The question of whether or not Thurston's construction gives a tiling when conditions are relaxed, is equivalent to a number of questions in different fields in mathematics and computer science, like spectral theory (see Siegel [Sie04]), the theory of quasicrystals (Arnoux et al [ABEI01]), discrete geometry (Ito and Rao [IR06]) and automata ([Sie04]). In [Aki02], Akiyama defined a weak finiteness property (W) and proved that it is equivalent to the tiling property.…”

Section: Introductionmentioning

confidence: 99%

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Kalle

Steiner

2012

Trans. Amer. Math. Soc.

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International audienceFrom the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base~$\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

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Section: Introductionmentioning

confidence: 99%

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Kalle

Steiner

2012

Trans. Amer. Math. Soc.

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“…The field of aperiodic order aims to study such patterns, and to establish connections between their properties, and their constructions, to other fields of mathematics and the natural sciences. To name a few, aperiodic order has interactions with areas of mathematics such as mathematical logic [29]-as established by Berger's proof of the undecidability of the domino problem [8], Diophantine approximation [2,9,20,21], the structure of attractors [12] and symbolic dynamics [40], and notably is of relevance to solid state physics in the wake of the discovery of quasicrystals by Shechtman et al [41].…”

Section: Introductionmentioning

confidence: 99%

Pattern-equivariant homology

Walton

2017

Algebr. Geom. Topol.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Pattern-equivariant homology JAMES J WALTONPattern-equivariant (PE) cohomology is a well-established tool with which to interpret theČech cohomology groups of a tiling space in a highly geometric way. We consider homology groups of PE infinite chains and establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example; we show how through this formalism one may give highly approachable geometric descriptions of the generators of theČech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincaré duality fails over integer coefficients for the "ePE homology groups" based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.52C23; 37B50, 52C22, 55N05 IntroductionIn the past few decades a rich class of highly ordered patterns has emerged whose central examples, despite lacking global translational symmetries, exhibit intricate internal structure, imbuing these patterns with properties akin to those enjoyed by periodically repeating patterns. The field of aperiodic order aims to study such patterns, and to establish connections between their properties, and their constructions, to other fields of mathematics and the natural sciences. PE cohomology allows for an intuitive description of theČech cohomology L H . / of tiling spaces. Over R coefficients the PE cochain groups may be defined using PE differential forms [24], and over general abelian coefficients, when the tiling has a cellular structure, with PE cellular cochains; see Sadun [37]. Rather than just providing a reflection of topological invariants of tiling spaces, on the contrary, these PE invariants are of principal relevance to aperiodic structures and their connections with other fields in their own right; see, for example, Kelly and Sadun's use of them [29] in a topological proof of theorems of Kesten and Oren regarding the discrepancy of irrational rotations. It is perhaps more appropriate to view the isomorphism between L H . / and the PE cohomology as an elegant interpretation of the PE cohomology, rather than vice versa, of theoretical and computational importance.In this paper we introduce the pattern-equivariant homology gro...

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“…Discrete planes can be seen as an union of square faces. Such stepped surface, introduced in [IO94, IO93] as a way to construct quasiperiodic tilings of the plane, can be generated from multidimensional continued fraction algorithms by introducing substitutions on square faces [ABEI01,ABI02].…”

Section: Introductionmentioning

confidence: 99%

A $$d$$ d -dimensional Extension of Christoffel Words

Labbé

Reutenauer

2015

Discrete Comput Geom

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In this article, we extend the definition of Christoffel words to directed subgraphs of the hypercubic lattice in arbitrary dimension that we call Christoffel graphs. Christoffel graphs when d = 2 correspond to well-known Christoffel words. Due to periodicity, the d-dimensional Christoffel graph can be embedded in a (d − 1)-torus (a parallelogram when d = 3). We show that Christoffel graphs have similar properties to those of Christoffel words: symmetry of their central part and conjugation with their reversal. Our main result extends Pirillo's theorem (characterization of Christoffel words which asserts that a word amb is a Christoffel word if and only if it is conjugate to bma) in arbitrary dimension. In the generalization, the map amb → bma is seen as a flip operation on graphs embedded in Z d and the conjugation is a translation. We show that a fully periodic subgraph of the hypercubic lattice is a translate of its flip if and only if it is a Christoffel graph.

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Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

Cited by 23 publications

References 21 publications

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Pattern-equivariant homology

A $$d$$ d -dimensional Extension of Christoffel Words

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