# Singular substitutions of constant length

## TL;DR

This paper investigates the spectral properties of primitive aperiodic substitutions of constant length, establishing a necessary condition involving eigenvalues for the presence of a Lebesgue spectral component.

## Contribution

It introduces a new spectral criterion linking eigenvalues of the substitution matrix to Lebesgue spectrum presence in substitution dynamical systems.

## Key findings

- A Lebesgue component requires an eigenvalue of magnitude √q.
- The proof combines Queffélec's results with local spectral measure estimates.
- Provides a necessary condition for spectral types in substitution systems.

## Abstract

We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals $\sqrt{q}$ in absolute value. The proof is based on results of M. Queff\'elec, combined with estimates of the local dimension of the spectral measure at zero.