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

TL;DR

This survey explores the spectral properties of primitive aperiodic substitution dynamical systems, focusing on the continuous spectrum and its representations via matrix Riesz products and spectral cocycles.

Contribution

It provides a comprehensive overview of the spectral analysis methods for substitution systems, highlighting the connections between spectral measures, Riesz products, and spectral cocycles.

Findings

Maximal spectral type for Z-actions via matrix Riesz products

Local dimension of spectral measure governed by spectral cocycle

Emphasis on ideas, connections, and references for proofs

Abstract

In 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.