Markov partitions for toral ${ℤ}^{2}$-rotations featuring Jeandel–Rao Wang shift and model sets
Annales Henri Lebesgue, Volume 4 (2021) , pp. 283-324.

KeywordsWang tilings, aperiodic, rotation, Markov partition, cut and project

### Abstract

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 $\left\{1,2,8\right\}$. 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 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.

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