Papers › Markov partitions for toral ℤ²-rotations featuring Jeandel-Rao Wang shift and model sets
Markov partitions for toral ℤ²-rotations featuring Jeandel-Rao Wang shift and model sets
Sébastien Labbé
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We define a partition 𝒫₀ and a ℤ²-rotation (ℤ²-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 ℤ²-rotation on 𝕋² 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 ℤ²-rotation on 𝕋². We prove in both cases that the toral ℤ²-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 ℤ²-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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