A self-similar aperiodic set of 19 Wang tiles

Labbé, Sébastien

doi:10.1007/s10711-018-0384-8

Cited by 12 publications

(39 citation statements)

References 41 publications

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“…(This result was proven in [14] for a tile set τ constructed in [12]. Other examples of minimal aperiodic SFTs were suggested in in [30,29].) Our next result in some sense strengthens Theorem 2: we claim that there exists a quasiperiodic SFT that contains only non-computable configurations.…”

Section: Quasiperiodicity Is Compatible With Non-computabilitysupporting

confidence: 69%

The expressiveness of quasiperiodic and minimal shifts of finite type

Durand

Romashchenko

2020

Ergod. Th. Dynam. Sys.

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We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for the effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. Supported by the ANR grant Racaf ANR-15-CE40-0016-01. Preliminary versions of some of the presented results we published in conference papers on MFCS-2015 [23] (Theorems 3-4) and MFCS-2017 [26] (Theorems 6-7).

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Section: Quasiperiodicity Is Compatible With Non-computabilitysupporting

confidence: 69%

The expressiveness of quasiperiodic and minimal shifts of finite type

Durand

Romashchenko

2020

Ergod. Th. Dynam. Sys.

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“…The self-similarity of Ω U is given by the following 2-dimensional morphism: [Lab18] that Ω U is self-similar, aperiodic and minimal.…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…The square of the morphism ω U has eight distinct fixed points z = ω U (z) ∈ Ω U that can be generated from the following values of z at the origin: Proof. The proof is done by applying the morphism ω U on each of the fifty 2 × 2 factors of the language of Ω U already computed in [Lab18]. We get a graph shown in Figure 12 Finally we illustrate in Figure 13 the language of horizontal dominoes in Ω U as a Rauzy graph.…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…If a Wang tile set T has a subset of markers in the direction e i then there exists a radius r ∈ N such that FindMarkers(T , i, r) from Algorithm 1 outputs it. We can now state a small generalization of Theorem 10 from [Lab18]. Indeed, while the presence of markers allows to desubstitute tilings, there is a choice to be made.…”

Section: Definition 12 Let T Be a Wang Tile Set And Let ω T Be Its Wmentioning

confidence: 99%

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Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Labbé

2019

Discrete Comput Geom

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We describe the substitutive structure of Jeandel-Rao aperiodic Wang tilings Ω 0 . We introduce twelve sets of Wang tiles {T i } 1≤i≤12 together with their associated Wang shifts {Ω i } 1≤i≤12 . Using a method proposed in earlier work, we prove the existence of recognizable 2dimensional morphisms ω i : Ω i+1 → Ω i for every i ∈ {0, 1, 2, 3, 6, 7, 8, 9, 10, 11} that are onto up to a shift. Each ω i maps a tile on a tile or on a domino of two tiles. We also prove the existence of a topological conjugacy η : Ω 6 → Ω 5 which shears Wang tilings by the action of the matrix ( 1 1 0 1 ) and an embedding π : Ω 5 → Ω 4 that is unfortunately not onto. The Wang shift Ω 12 is self-similar, aperiodic and minimal. Thus we give the substitutive structure of a minimal aperiodic Wang subshift X 0 of the Jeandel-Rao tilings Ω 0 . The subshift X 0 Ω 0 is proper due to some horizontal fracture of 0's or 1's in tilings in Ω 0 and we believe that Ω 0 \ X 0 is a null set. Algorithms are provided to find markers, recognizable substitutions and shearing topological conjugacy from a set of Wang tiles.

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“…The study of bidimensional words is less developed, even though many concepts and results are naturally extendable from the unidimensional case (see e.g. [2,5,6,10,14,15,22,26]). However, some words problems become much more difficult in dimensions higher than one.…”

Section: Introductionmentioning

confidence: 99%

Recurrence in Multidimensional Words

Charlier

Puzynina

Vandomme

2019

Language and Automata Theory and Applications

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In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all ps 1 , . . . , s d q P N d there exists n P N such that each block of size pn, . . . , nq contains the prefix of size ps 1 , . . . , s d q.We are interested in a modification of this property. Namely, we ask that for each rational direction pq 1 , . . . , q d q, each rectangular prefix occurs along this direction in positions pq 1 , . . . , q d q with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional words satisfying this condition, and more generally, a series of four increasingly stronger conditions. In particular, we study the uniform recurrence along directions of multidimentional rotation words and of fixed points of square morphisms.

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A self-similar aperiodic set of 19 Wang tiles

Cited by 12 publications

References 41 publications

The expressiveness of quasiperiodic and minimal shifts of finite type

The expressiveness of quasiperiodic and minimal shifts of finite type

Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Recurrence in Multidimensional Words

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