Spectral properties of substitutions on compact alphabets

TL;DR

This paper extends the spectral theory of substitutions to compact and infinite alphabets, providing conditions for different spectral types, and constructing examples with complex spectral components.

Contribution

It introduces new spectral conditions for substitutions on compact alphabets and presents the first example with infinite Lebesgue and singular continuous spectral components.

Findings

Provided sufficient conditions for pure point, absolutely continuous, and singular continuous diffraction.

Constructed examples with countably infinite Lebesgue and singular continuous spectral components.

Extended spectral theory to substitutions on infinite alphabets and Delone sets of infinite type.

Abstract

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non-constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.