Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, 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 T_0 of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal 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 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 stepswhere 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}^2-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions  and shear conjugacy from a set of Wang tiles.
Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane , none of them being invariant under a nontrivial translation. We study herein the Wang shift \Omega_0 made of all valid tilings using the set T_0 of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift X_0 of \Omega_0 such that every 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 19 self-similar and aperiodic Wang tiles. We suggest that this provides an almost complete description of the substitutive structure of Jeandel-Rao 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 12 elementary steps, 10 of which involve the same procedure allowing one to desubstitute Wang tilings from the existence of a subset of marker tiles. The 2 other steps involve the addition of decorations to deal with fault lines and changing the base of the \mathbb{Z}^2-action through a shear conjugacy. Algorithms are provided to find markers, recognizable substitutions , and shear conjugacy from a set of Wang tiles.
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