Markov partitions for toral 2 -rotations featuring Jeandel–Rao Wang shift and model sets
Annales Henri Lebesgue, Volume 4 (2021) , pp. 283-324.

Metadata

KeywordsWang tilings, aperiodic, rotation, Markov partition, cut and project

Abstract

We define a partition 𝒫 0 and a 2 -rotation ( 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 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 2 -rotation on 𝕋 2 . We prove in both cases that the toral 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 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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