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Spectral theory of \(\mathbb{Z}^{d}\) substitutions. (English) Zbl 1388.37021

Summary: In this paper, we generalize and develop results of M. Queffélec [Substitution dynamical systems. Spectral analysis. 2nd ed. Dordrecht: Springer (2010; Zbl 1225.11001)] allowing us to characterize the spectrum of an aperiodic \(\mathbb{Z}^{d}\) substitution. Specifically, we describe the Fourier coefficients of mutually singular measures of pure type giving rise to the maximal spectral type of the translation operator on \(L^{2}\), without any assumptions on primitivity or height, and show singularity for aperiodic bijective commutative \(\mathbb{Z}^{d}\) substitutions. Moreover, we provide a simple algorithm to determine the spectrum of aperiodic \(\mathbf{q}\)-substitutions, and use this to show singularity of Queffélec’s non-commutative bijective substitution, as well as the Table tiling, answering an open question of B. Solomyak [Discrete Comput. Geom. 20, No. 2, 265–279 (1998; Zbl 0919.52017)]. Finally, we show that every ergodic matrix of measures on a compact metric space can be diagonalized, which we use in the proof of the main result.

MSC:

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37A30 Ergodic theorems, spectral theory, Markov operators
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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