Substitutive structure of Jeandel-Rao aperiodic tilings
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We describe the substitutive structure of Jeandel-Rao aperiodic Wang tilings Ω_0. We introduce twelve sets of Wang tiles {T_i}_1≤ i≤ 12 together with their associated Wang shifts {Ω_i}_1≤ i≤ 12. Using a method proposed in earlier work, we prove the existence of recognizable 2-dimensional morphisms ω_i:Ω_i+1→Ω_i for every i∈{0,1,2,3,6,7,8,9,10,11} that are onto up to a shift. Each ω_i maps a tile on a tile or on a domino of two tiles. We also prove the existence of a topological conjugacy η:Ω_6→Ω_5 which shears Wang tilings by the action of the matrix (< s m a l l m a t r i x >) and an embedding π:Ω_5→Ω_4 that is unfortunately not onto. The Wang shift Ω_12 is self-similar, aperiodic and minimal. Thus we give the substitutive structure of a minimal aperiodic Wang subshift X_0 of the Jeandel-Rao tilings Ω_0. The subshift X_0⊊Ω_0 is proper due to some horizontal fracture of 0's or 1's in tilings in Ω_0 and we believe that Ω_0∖ X_0 is a null set. Algorithms are provided to find markers, recognizable substitutions and sheering topological conjugacy from a set of Wang tiles.
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