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

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.