A minimal subsystem of the Kari–Culik tilings

Siefken, Jason

doi:10.1017/etds.2015.118

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…This is an analogue of a result known for Kari and Culik aperiodic Wang tilings which satisfy equations involving balanced representations of real numbers and orbits of piecewise rationally multiplicative maps, see also Theorem 16 in [ENP07] and Proposition 3 in [Sie17]. Here the result applies to all of the configurations in the Wang shift Ω n .…”

Section: Theorem Bsupporting

confidence: 53%

“…This construction reminds of the proof of existence of tilings with Kari and Culik tiles based on the balanced representation of real numbers and first difference of Beatty sequences [Kar96,Cul96], see also [ENP07,Sie17].…”

Section: Theorem Bmentioning

confidence: 75%

“…Based on the same idea, Culik [Cul96] proposed a smaller aperiodic set of 13 tiles (four tiles satisfying (1.1) with q = 3 and nine tiles with q = 1 2 ). Note that generalizations of Kari-Culik tilings exist [ENP07] and that further results were obtained about their entropy [DGG17] and on a minimal subsystem [Sie17].…”

Section: Introductionmentioning

confidence: 76%

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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: Theorem Bsupporting

confidence: 53%

Section: Theorem Bmentioning

confidence: 75%

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A self-similar aperiodic set of 19 Wang tiles

Labbé

2018

Geom Dedicata

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“…We prove the same result for the Wang shift Ω U (see Theorem 14.1). As opposed to the Kari-Culik Wang shift, for which a minimal subsystem is related to a dynamical system on p-adic numbers [Sie17], windows used for the cut and project schemes are Euclidean.…”

Section: Annales Henri Lebesguementioning

confidence: 99%

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

Labbé¹

2021

Annales Henri Lebesgue

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We define a partition P 0 and a Z 2 -rotation (Z 2 -action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition P U and a Z 2 -rotation on T 2 whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that P U is a Markov partition for the Z 2 -rotation on T 2 . We prove in both cases that the toral Z 2 -rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is {1, 2, 8}. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral Z 2 -rotations. It provides a construction

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“…We prove the same result for the Wang shift Ω U (see Theorem 4). As opposed to the Kari-Culik Wang shift, for which a minimal subsystem is related to a dynamical system on p-adic numbers [Sie17], windows used for the cut and project schemes are Euclidean.…”

Section: Introductionmentioning

confidence: 99%

Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets

Labbé

2019

Preprint

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We define a partition P 0 and a Z 2 -action by rotations on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition P U and Z 2 -action by rotations on T 2 whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that P U is a Markov partition for the Z 2 -action by rotations on T 2 . We prove in both cases that the Z 2 -action on the torus is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is {1, 2, 8}. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the Z 2 -action on the torus. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.Résumé. Nous définissons une partition P 0 et une Z 2 -action par rotations sur un tore 2dimensionnel dont le système dynamique symbolique associé est un sous-shift propre et minimal du sous-shift apériodique de Jeandel-Rao décrit par un ensemble de 11 tuiles de Wang. Nous définissons une autre partition P U et une Z 2 -action par rotations sur T 2 dont le système dynamique symbolique associé est égal au sous-shift minimal et apériodique défini par un ensemble de 19 tuiles de Wang. On montre que P U est une partition de Markov pour la Z 2 -action par rotations sur T 2 . Nous prouvons dans les deux cas que la Z 2 -action sur le tore est le facteur équicontinu maximal des sous-shifts minimaux et que l'ensemble des cardinalités des fibres du facteur est {1, 2, 8}. Les deux sous-shifts minimaux sont uniquement ergodiques et sont isomorphes en tant que systèmes dynamiques mesurés à la Z 2 -action sur le tore. Les résultats fournissent une construction des sous-shifts de Wang en tant qu'ensembles modèles par la méthode de coupe et projection 4 sur 2. Un puzzle à faire soi-même est disponible en annexe pour illustrer les résultats.

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A minimal subsystem of the Kari–Culik tilings

Cited by 5 publications

References 7 publications

A self-similar aperiodic set of 19 Wang tiles

A self-similar aperiodic set of 19 Wang tiles

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

Markov partitions for toral $\mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets

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