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

This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: Z \mathbb {Z} -actions and R \mathbb {R} -actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For Z \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. References are given to complete proofs and emphasize ideas and various links.