Article contents
Spectral theory of
$\mathbb{Z}^{d}$ substitutions
Published online by Cambridge University Press: 20 October 2016
Abstract
In this paper, we generalize and develop results of Queffélec 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 Solomyak. 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.
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- © Cambridge University Press, 2016
References
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