Spectral Theory of Z^d Substitutions

Abstract

In this paper, we generalize and develop results of Queffelec allowing us to characterize the spectrum of an aperiodic substitution in Z^d by describing the Fourier coefficients of mutually singular measures of pure type giving rise to the maximal spectral type of the translation operator on L^2. This is done without any assumptions on primitivity or height, and provides a simple algorithm for determining singularity to Lebesgue spectrum for such substitutions, and we use this to show singularity of the spectrum for Queffelec's noncommutative bijective substitution, as well as the Table tiling, answering an open question of Solomyak. Moreover, we also prove that the spectrum of any aperiodic bijective commutative Z^d substitution on a finite alphabet is purely singular. 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.