Matrix Riesz products

Abstract

Turning back to the correlation matrix ∑ = (σ αβ) associated, in the previous chapter, with the primitive and aperiodic substitution ζ of length q, we shall prove that ∑ is the weak-star limit point of a product of matrices whose entries are trigonometric polynomials, in a way similar to the case of generalized Riesz products. This provides us with a constructive process to explicit ∑ for special substitutions, such as commutative ones (Thue-Morse) but also for the Rudin-Shapiro substitution, and therefore, we will be able to deduce their maximal spectral type. © 2010 Springer-Verlag Berlin Heidelberg.