Markov partitions for toral <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:math>-rotations featuring Jeandel–Rao Wang shift and model sets

Labbé, Sébastien

doi:10.5802/ahl.73

Cited by 6 publications

(26 citation statements)

References 51 publications

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“…In the spirit of [39, Prop. 6.5.8] for Z-actions, we have the following proposition whose proof can be found in [35]. PROPOSITION 5.4.…”

Section: Factor Mapmentioning

confidence: 89%

“…PROPOSITION 5.4. [35,Prop. 5.1] Let P give a symbolic representation of the dynamical system (T, Z 2 , R).…”

Section: Factor Mapmentioning

confidence: 99%

“…Of course, Lemma 5.5, whose proof can also be found in [35], does not hold if P does not give a symbolic representation of (T, Z 2 , R). For example, consider the Z 2 -rotation on T 2 defined by R(n, x) = x + (n 1 α, n 2 β) for every n = (n 1 , n 2 ) ∈ Z 2 for some fixed irrational α, β ∈ R Q.…”

Section: Factor Mapmentioning

confidence: 99%

“…that are onto up to a shift, i.e., X P 8 ,R 8 = β 8 (X P 9 ,R 9 ) σ , X P 9 ,R 9 = β 9 (X P 10 ,R 10 ) σ and X P 10 ,R 10 = τ(X P 8 ,R 8 ) and the product β 8 β 9 τ is an expansive and primitive self-similarity. (v) The subshift X P 8 ,R 8 is topologically conjugate to the subshift X P U ,R U introduced in [35] as there exists a bijection ζ :…”

Section: Introductionmentioning

confidence: 99%

“…Part II defines the induction of toral Z 2 -rotations, induced toral partitions and renormalization schemes. In Part III, we induce the partition P 0 introduced in [35] to get a sequence of substitutions until we reach a self-induced partition. We prove that the substitutive structure obtained from the induction procedure is the same as the substitutive structure computed directly from the Jeandel-Rao Wang tiles in [36].…”

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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“…In the spirit of [39, Prop. 6.5.8] for Z-actions, we have the following proposition whose proof can be found in [35]. PROPOSITION 5.4.…”

Section: Factor Mapmentioning

confidence: 89%

“…PROPOSITION 5.4. [35,Prop. 5.1] Let P give a symbolic representation of the dynamical system (T, Z 2 , R).…”

Section: Factor Mapmentioning

confidence: 99%

Section: Factor Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Aubrun¹,

Bitar²,

Huriot-Tattegrain³

2023

Ergod. Th. Dynam. Sys.

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We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$ . In the case of $\mathbb {F}_n\times \mathbb {Z}$ , the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

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Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

BARBIERI,

LABBÉ

2024

Ergod. Th. Dynam. Sys.

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Two asymptotic configurations on a full $\mathbb {Z}^d$ -shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$ . We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$ . Many open questions are raised by the current work and are listed in the introduction.

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Markov partitions for toral ℤ 2 -rotations featuring Jeandel–Rao Wang shift and model sets

Cited by 6 publications

References 51 publications

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

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

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Indistinguishable asymptotic pairs and multidimensional Sturmian configurations

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