Self-similarity and spectral theory: on the spectrum of substitutions

survey (no new results), dedicated to N. K. Nikolski; added in this version: a small correction in the historical introduction; a remark on the convergence of generalized Riesz products at the end of Section 3In this survey of the spectral properties of substitution dynamical systems we consider primitive aperiodic substitutions and associated dynamical systems: ${\mathbb Z}$-actions and ${\mathbb R}$-actions, the latter viewed as tiling flows. Our focus is on the continuous part of the spectrum. For ${\mathbb Z}$-actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. We give references to complete proofs and emphasize ideas and connections