The centralizer (automorphism group) and normalizer (extended symmetry group) of the shift action inside the group of self-homeomorphisms are studied, in the context of certain $ \mathbb{Z}^d $ subshifts with a hierarchical supertile structure, such as bijective substitutive subshifts and the Robinson tiling. Restrictions on these groups via geometrical considerations are used to characterize explicitly their structure: nontrivial extended symmetries can always be described via relabeling maps and rigid transformations of the Euclidean plane permuting the coordinate axes. The techniques used also carry over to the well-known Robinson tiling, both in its minimal and non-minimal versions.
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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 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 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} $
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An example of applying a rectangular substitution to a pattern
Points from the two-dimensional Thue-Morse substitution. The first two configurations correspond to (the central pattern of) two points
In the figure, we see how
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
The five types of Robinson tiles, resulting in an alphabet of
The formation of a second order supertile of size
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
Two possible ways in which the tiling from Figure 8 exhibits fracture-like behavior, resulting in valid points from
The substructure of a point of
How a shift by
The relabeling map