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Spectral theory of spin substitutions

  • *Corresponding author: Neil Mañibo

    *Corresponding author: Neil Mañibo

Dedicated to our late friend and colleague, Uwe Grimm
The second author is funded by the German Research Foundation (DFG, Deutsche Forschungs-gemeinschaft), via SFB 1283/2 2021-317210226

Abstract / Introduction Full Text(HTML) Figure(10) / Table(1) Related Papers Cited by
  • We introduce substitutions in $ {\mathbb{Z}}^m $ which have non-rectangular domains based on an endomorphism $ Q $ of $ {\mathbb{Z}}^m $ and a set $ {\mathcal D} $ of coset representatives of $ {\mathbb{Z}}^m/Q{\mathbb{Z}}^m $, which we call digit substitutions. Using a finite abelian 'spin' group we define 'spin digit substitutions' and their subshifts $ ({\Sigma}, {\mathbb{Z}}^m) $. Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an $ m $-dimensional odometer are given, inducing a complete decomposition of the function space $ L^{2}({\Sigma},\mu) $. This enables the use of group characters in $ {\widehat{G}} $ to derive substitutive factors and analyze the spectra of specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous, and singular continuous spectral measures, together with some bounds on their spectral multiplicity.

    Mathematics Subject Classification: Primary: 37B10, 37A30; Secondary: 37B52, 52C23, 43A25.

    Citation:

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  • 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 $ ($\textsf{ac}$)

    Table 1.  Numerical values for upper bounds for $ 2f(N) $ for $ B_{\chi^{ }_1} $. Here $ \|\cdot\|^{ }_{\text{F}} $ stands for the Frobenius norm. All numerical errors are less than $ 10^{-3} $

    $ N $ 10 11 12 13
    $ \frac{1}{N}\int_{\mathbb{T}}\log\|B_{\chi^{ }_1}^{(N)}(\vec{x})\|^{2}_{\text{F}} $ 0.703953 0.695342 0.688005 0.682035
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