Extended symmetry groups of multidimensional subshifts with hierarchical structure

Figure 1.  An example of applying a rectangular substitution to a pattern

Figure 4.  Points from the two-dimensional Thue-Morse substitution. The first two configurations correspond to (the central pattern of) two points $ x, y\in\mathsf{X}_{\theta_{\text TM}} $ matching exactly in one half-plane, as in Lemma 4.4. The third configuration is an "illegal" point $ z\in\mathsf{X}_{\theta_{\text TM}}^*\setminus\mathsf{X}_{\theta_{\text TM}} $ from the extended substitutive subshift. The associated seeds and substitution rule are shown below

Figure 2.  $ 2^n\times 2^n $ grids associated with the iterates of a primitive substitution $ \theta $ in a point from a substitutive subshift. The corresponding substitution is indicated on the right

Figure 3.  In the figure, we see how $ x|_{K_{\boldsymbol{p}}} = \theta^m(a) $ (for some $ a\in\mathcal{A} $) determines $ f(x)|_{K_{\boldsymbol{p}}^{\circ r}} $ and, in particular, $ f(x)|_{I_{\boldsymbol{p}}} $. Since the substitution is bijective, this forces $ f(x)|_{L_{\boldsymbol{p}}} $ to equal $ \theta^m(b) $ for some $ b\in\mathcal{A} $ which depends solely on $ a $

Figure 5.  The situation in the proof of Lemma 4.6. As the side length of the rectangles associated with the substitution increases exponentially, the inner product $ \langle\boldsymbol{v}, \boldsymbol{w}\rangle $ which determines whether $ \boldsymbol{w} $ belongs to $ S $ or $ S' $ (or neither) takes sufficiently many different (integer) values inside any of these rectangles to ensure that at least one such rectangle intersects both $ S $ and $ S' $

Figure 6.  The five types of Robinson tiles, resulting in an alphabet of $ 28 $ symbols after applying all possible rotations and reflections. The third tile is usually called a cross

Figure 7.  The formation of a second order supertile of size $ 3\times 3 $

Figure 8.  A fragment of a point from the Robinson shift, distinguishing the four supertiles involved, the vertical and horizontal strips of tiles separating each supertile and the $ 2\mathbb{Z}\times 2\mathbb{Z} $ sublattice that contains only crosses. Note that the tiles in the vertical strip separating the supertiles are copies of the first tile of Figure 6 with the same orientation

Figure 9.  Two possible ways in which the tiling from Figure 8 exhibits fracture-like behavior, resulting in valid points from $ X_{\text Rob} $

Figure 10.  The substructure of a point of $ X_{\text Rob} $ in terms of $ n $-th order supertiles. Note how all supertiles overlap either $ S^+ $ or $ S^- $

Figure 11.  How a shift by $ k_1\boldsymbol{q} $ makes the arrangement of supertiles in $ S^+ $ not match with the corresponding tiles in $ S^- $

Figure 12.  The relabeling map $ \mathfrak{R} $ which replaces each tile with its corresponding rotation by $ \frac{1}{2}\pi $