Singular substitutions of constant length

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. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.

Corrected typos and added remarks and references

References in corpus (1)

• Spectrum of a Rudin-Shapiro-like sequence

Cited by in corpus (4)

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• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Uniformly distributed orbits in ${\mathbb{T}}^d$ and singular substitution dynamical systems