Spectral theory of substitutions

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.