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

Labbé, Sébastien

Annales Henri Lebesgue, Tome 4 (2021), pp. 283-324.

Nous définissons une partition ${𝒫}_0$ et une ${\mathbb{Z}}^2$-rotation (${\mathbb{Z}}^2$-action définie par des rotations)sur un tore 2-dimensionnel dont le système dynamique symbolique associé est un sous-décalage propre et minimal du sous-décalage apériodique de Jeandel–Rao décrit par un ensemble de 11 tuiles de Wang. Nous définissons une autre partition ${𝒫}_{𝒰}$ et une ${\mathbb{Z}}^2$-rotation sur ${𝕋}^2$ dont le système dynamique symbolique associé est égal au sous-décalage minimal et apériodique défini par un ensemble de 19 tuiles de Wang. On montre que ${𝒫}_{𝒰}$ est une partition de Markov pour la ${\mathbb{Z}}^2$-rotation sur ${𝕋}^2$. Nous prouvons dans les deux cas que la ${\mathbb{Z}}^2$-rotation sur le tore est le facteur équicontinu maximal des sous-décalages minimaux et que l’ensemble des cardinalités des fibres du facteur est $\{1,2,8\}$. Les deux sous-décalages minimaux sont uniquement ergodiques et sont isomorphes en tant que systèmes dynamiques mesurés à la ${\mathbb{Z}}^2$-rotation sur le tore. Les résultats fournissent une construction de ces sous-décalages 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.

We define a partition ${𝒫}_0$ and a ${\mathbb{Z}}^2$-rotation (${\mathbb{Z}}^2$-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 ${\mathbb{Z}}^2$-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 ${\mathbb{Z}}^2$-rotation on ${𝕋}^2$. We prove in both cases that the toral ${\mathbb{Z}}^2$-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 ${\mathbb{Z}}^2$-rotations. 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.

Classification : 37B50, 52C23, 28D05, 37B05
Mots clés : Wang tilings, aperiodic, rotation, Markov partition, cut and project

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