Spectral theory of spin substitutions

Figure 1.  The supertiles $ \mathcal{S}^{1} ({\mathsf{a}}), \mathcal{S}^{2} ({\mathsf{a}}) $, and $ \mathcal{S}^{3} ({\mathsf{a}}) $ for Example 2

Figure 2.  The word $ (0,0),(0,-1), (-1,-1), (-1,0) \mapsto \, {\mathsf{a}},{\mathsf{a}},{\mathsf{b}},{\mathsf{a}} $ substituted six times. The presence of arbitrarily large rectangular words allows us to define a subshift of $ {\mathbb{Z}}^2 $ for $ {\mathcal{S}} $

Figure 3.  Three iterations of the pink $ {\mathsf{a}} $-tile located at the origin

Figure 4.  The image of some $ {\tau} \in {\Sigma} $ embedded into the tiling spaces $ {\Omega}_{{\mathfrak{u}}} $ (top) and $ {\Omega}_{{\mathfrak{t}}} $ (bottom)

Figure 5.  The alphabet. Spins are depicted as colors that vary in shade by digit

Figure 6.  The 1- and 2-supertiles for the triomino substitution, arranged by digit and spin as in Figure 5

Figure 7.  An indication that spins are uniformly distributed in the triomino subshift

Figure 8.  The level -1 and -2 supertiles of the Vierdrachen substitution

Figure 9.  The level-9 supertile for $ {\mathsf{d}}_0 $ and its image under the forget-the-spins and the forget-the-digits maps

Figure 10.  The image of $ {\mathcal{S}}^{9}({\mathsf{d}}_0) $ under the single-block codes given by the nontrivial characters. From (L) to (R), the corresponding character $ \chi $ and the spectral type of $ H^{\chi} $: $ \chi^{ }_1 $ ($\textsf{ac}$), $ \chi^{ }_2 $ ($\textsf{sc}$) and $ \chi^{ }_3 $ ($\textsf{ac}$)