Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Labbé, Sébastien

doi:10.1007/s00454-019-00153-3

Cited by 11 publications

(33 citation statements)

References 26 publications

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“…The tile set U comes from the study of the structure of Jeandel-Rao aperiodic tilings. The link with Jeandel-Rao tilings needs more tools and space and will be done in a forthcoming paper [Lab18b]. In this contribution, we prove the following result.…”

Section: Introductionmentioning

confidence: 69%

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A self-similar aperiodic set of 19 Wang tiles

Labbé

2018

Geom Dedicata

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We define a Wang tile set U of cardinality 19 and show that the set Ω U of all valid Wang tilings Z 2 → U is self-similar, aperiodic and is a minimal subshift of U Z 2 . Thus U is the second smallest self-similar aperiodic Wang tile set known after Ammann's set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable 2-dimensional morphism ω : Ω U → Ω U that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.

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Section: Introductionmentioning

confidence: 69%

“…Each step has the effect of identifying special rows (or columns) made of marker tiles (see Definition 9) that have to be adjacent in one direction but can not be adjacent in the other direction. The wider applicability of this method (Theorem 10) allowing to be automated by computer check will be used in [Lab18b].…”

Section: Authormentioning

confidence: 99%

A self-similar aperiodic set of 19 Wang tiles

Labbé

2018

Geom Dedicata

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“…Theorem 1.2 must be compared with the main result of [36], recalled herein as Theorem 13.1, giving the substitutive structure of a minimal subshift X 0 of the Jeandel-Rao Wang shift Ω 0 . That result proved the existence of sets of Wang tiles {T i } 1≤i ≤12 together with their associated Wang shifts {Ω i } 1≤i ≤12 and 2-dimensional morphisms ω i : Ω i +1 → Ω i that provide the substitutive structure of Jeandel-Rao Wang shift, see Figure 4.…”

Section: Introductionmentioning

confidence: 99%

“…We prove that the subshifts X 0 ⊂ Ω 0 and X P 0 ,R 0 are equal since they have a common substitutive structure. The substitutive structure of X 0 computed in [36] and the substitutive structure of X P 0 ,R 0 satisfy…”

Section: Introductionmentioning

confidence: 99%

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Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

Labbé¹

2021

JMD

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<p style='text-indent:20px;'>We extend the notion of Rauzy induction of interval exchange transformations to the case of toral <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation, i.e., <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action defined by rotations on a 2-torus. If <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> denotes the symbolic dynamical system corresponding to a partition <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{P} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> is Cartesian on a sub-domain <inline-formula><tex-math id="M8">\begin{document}$ W $\end{document}</tex-math></inline-formula>, we express the 2-dimensional configurations in <inline-formula><tex-math id="M9">\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}</tex-math></inline-formula> as the image under a <inline-formula><tex-math id="M10">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-dimensional morphism (up to a shift) of a configuration in <inline-formula><tex-math id="M11">\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M12">\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}</tex-math></inline-formula> is the induced partition and <inline-formula><tex-math id="M13">\begin{document}$ \widehat{R}|_W $\end{document}</tex-math></inline-formula> is the induced <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-action on <inline-formula><tex-math id="M15">\begin{document}$ W $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We focus on one example, <inline-formula><tex-math id="M16">\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}</tex-math></inline-formula>, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift <inline-formula><tex-math id="M17">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr{P}}_0 $\end{document}</tex-math></inline-formula> is a Markov partition for the associated toral <inline-formula><tex-math id="M19">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M20">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>. It also implies that the subshift <inline-formula><tex-math id="M21">\begin{document}$ X_0 $\end{document}</tex-math></inline-formula> is uniquely ergodic and is isomorphic to the toral <inline-formula><tex-math id="M22">\begin{document}$ \mathbb{Z}^2 $\end{document}</tex-math></inline-formula>-rotation <inline-formula><tex-math id="M23">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and code to reproduce the proofs are provided.</p>

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A Numeration System for Fibonacci-Like Wang Shifts

Labbé¹,

Lepšová²

2021

Lecture Notes in Computer Science

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Substitutive Structure of Jeandel–Rao Aperiodic Tilings

Cited by 11 publications

References 26 publications

A self-similar aperiodic set of 19 Wang tiles

A self-similar aperiodic set of 19 Wang tiles

Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations

A Numeration System for Fibonacci-Like Wang Shifts

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