We describe the substitutive structure of Jeandel-Rao aperiodic Wang tilings \Omega_0. We introduce twelve sets of Wang tiles \{\mathcal{T_{i}\}_{1\leq i\leq 12} together with their associated Wang shifts \{i\}_{1\leq i\leq 12}. Using a method proposed in earlier work, we prove the existence of recognizable 2-dimensional morphisms \omega_i:i+1}%DIFDELCMD < \to%%% \Omega_i for every i\in\{0, 1, 2 , 3,6,7,8,9,10,11\} that are onto up to a shift. Each \omega_i maps a tile on a tile or on a domino of two tiles . We also prove the existence of a topological conjugacy \eta:\Omega_6%DIFDELCMD < \to%%% \Omega_5 which shears Wang tilings by the action of the matrix%DIFDELCMD < \left(smallmatrix 1 & 1\\ 0 & 1 smallmatrix\right) %%% and an embedding \pi:\Omega_5%DIFDELCMD < \to%%% \Omega_4 that is unfortunately not onto. The Wang shift \Omega _{12 subshift X_0 of the Jeandel-Rao tilings \Omega_0 . The subshift X_0%DIFDELCMD < \subsetneq%%% \Omega_0 is proper due to some horizontal fracture of 0's or 1's in tilings in \Omega_0 and we believe that \Omega_0\setminus X_0 is a null set . Algorithms are provided to find markers, recognizable substitutions and sheering topological conjugacy from a set of Wang tiles.
Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles _{i}\}_{1\leq i\leq 12} together with their associated Wang shifts \{i\}_{1\leq i\leq 12}. Using a method proposed in earlier work, we prove the existence of recognizable 2-dimensional morphisms \omega_i:i+1}%DIFDELCMD < \to%%% , %DIFDELCMD < \to%%% %DIFDELCMD < \left(smallmatrix 1 & 1\\ 0 & 1 smallmatrix\right) %%% %DIFDELCMD < \to%%% i.e., unit squares with colored edges, that admit valid (contiguous edges of adjacent tiles have the same color) tilings of the plane none of them being invariant under a nontrivial translation. In this contribution, we study the Wang shift \Omega _0 made of all valid tilings using the set \mathcal{T subshift X_0 of \Omega_0 such that any tiling in X_0 can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112 that are equivalent to a set of self-similar and aperiodic 19 Wang tiles. We think that this gives an almost complete description of the substitutive structure of Jeandel-Rao %DIFDELCMD < \subsetneq%%% tilings as we believe that \Omega_0\setminus X_0 is a null set for any shift-invariant probability measure on \Omega_0. The proof is based on twelve elementary steps where ten of them are using the same procedure allowing to desubstitute Wang tilings from the existence of a subset of marker tiles. The two other steps are adding decorations to deal with fault lines and are changing the base of the \mathbb{Z Algorithms are provided to find markers, recognizable substitutions and shear conjugacy from a set of Wang tiles.
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