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تسجيل مستخدم جديدMarkov partitions for toral $mathbb{Z}^2$-rotations featuring Jeandel-Rao Wang shift and model sets
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تأليف
Sebastien Labbe
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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.
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