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

Labbé, Sébastien

doi:10.5802/ahl.73

Labbé, Sébastien

Markov partitions for toral

ℤ^{2}

-rotations featuring Jeandel–Rao Wang shift and model sets

Annales Henri Lebesgue, Volume 4 (2021), pp. 283-324.

Metadata

DOI10.5802/ahl.73

Keywords Wang 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 ${1, 2, 8}$ . 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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$Markov partitions for toral <span class="mathjax-formula" data-tex="$\protect \mathbb{Z}^2$"><math xmlns="http://www.w3.org/1998/Math/MathML" ><msup><mi>ℤ</mi> <mn>2</mn> </msup></math></span>-rotations featuring Jeandel–Rao Wang shift and model sets - Annales Henri Lebesgue$

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