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.

v1:26 pages, 10 figures. v2:32 pages, 13 figures. v3: various small fixes