Abstract
We define a partition $\mathcal{P}_0$ 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 $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$. 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.
Highlights
While a tiling by a set of Wang tiles T is a tiling of the plane R2 whose validity is preserved by translations of R2, w6e6pr6efe6r t6o c6on6sid6er6m6ap6s Z62 →6 6T,6 th6at[6] w6e call configurations, whose validity is p5res4er7ved[5] b4y 7tra5nsl4at7ion5s o4f Z7 2.5 T4he7se5t Ω4T 7of all valid configurations Z2 → T is called a Wang shift as it is closed under the shift σ by integer translates
It was shown that the action of R2 by translation on the set of Penrose tilings is an almost one-to-one extension of a minimal R2-action by rotations on T4 [Rob96]
We do not know if Penrose tilings can be seen as a symbolic dynamical system associated to a Markov partition like it is the case for the Jeandel–Rao Wang shift
Summary
We consider Wang tilings from the point of view of symbolic dynamics [Rob04]. It was shown that the action of R2 by translation on the set of Penrose tilings is an almost one-to-one extension of a minimal R2-action by rotations on T4 [Rob96] (the fact that it is T4 instead of T2 is related to the consideration of tilings instead of shifts) This result can be seen as a higher dimensional generalization of the Sturmian dynamical systems. Note that a shift of finite type or Wang shift can be explicitly constructed from the Penrose tiling dynamical system, as shown in [SW03] This calls for a common point of view including Jeandel–Rao aperiodic tilings, Penrose tilings and others. We do not know if Penrose tilings can be seen as a symbolic dynamical system associated to a Markov partition like it is the case for the Jeandel–Rao Wang shift. It is possible that such Markov partitions exist only for tilings associated to some algebraic numbers, see [BF20]
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