Substitution sequences in \mathbb{Z}^{d} with a non-simple Lebesgue component in the spectrum

TLDR: In this paper, the authors construct d-dimensional substitution sequences for which the continuous part of the spectrum is generated by measures equal to Lebesgue measure, and show that random application of these substitution sequences produces sequences with a lebesgue component in their spectrum.

Abstract: We construct d-dimensional substitution sequences for which the continuous part of the spectrum is generated by measures equal to Lebesgue measure. A special case is the Rudin–Shapiro substitution sequence. The construction uses Hadamard matrices in an essential way, so the dimension and size of a substitution is restricted by the size of the Hadamard matrix defining it. Each such substitution automatically has a dual substitution, which is defined by the same Hadamard matrix, and which retains a Lebesgue spectral component. We also see that random application of our substitutions produces sequences with a Lebesgue component in their spectrum. Finally, we see that any d-dimensional substitution with d > 1 can be ‘unraveled’ into lower-dimensional substitutions which still have Lebesgue spectral components.