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# Extended symmetry groups of multidimensional subshifts with hierarchical structure

The author is supported by ANID-PFCHA/Doctorado Nacional/2017-21171061 (formerly CONICYT). Please check the Acknowledgments section below for further details

• 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.

Mathematics Subject Classification: 37B10, 37B51, 20B27, 52C23.

 Citation:

• 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 $\frac{1}{2}\pi$

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