Spectral theory of spin substitutions
Natalie Priebe Frank, Neil Mañibo · Discrete and Continuous Dynamical Systems · 2022
We introduce substitutions in \begin{document}$ {\mathbb{Z}}^m $\end{document} which have non-rectangular domains based on an endomorphism \begin{document}$ Q $\end{document} of \begin{document}$ {\mathbb{Z}}^m $\end{document} and a set \begin{document}$ {\mathcal D} $\end{document} of coset representatives of \begin{document}$ {\mathbb{Z}}^m/Q{\mathbb{Z}}^m $\end{document} , which we call digit substitutions. Using a finite abelian 'spin' group we define 'spin digit substitutions' and their subshifts \begin{document}$ ({\Sigma}, {\mathbb{Z}}^m) $\end{document} . Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an \begin{document}$ m $\end{document} -dimensional odometer are given, inducing a complete decomposition of the function space \begin{document}$ L^{2}({\Sigma},\mu) $\end{document} . This enables the use of group characters in \begin{document}$ {\widehat{G}} $\end{document} to derive substitutive factors and analyze the spectra of specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous, and singular continuous spectral measures, together with some bounds on their spectral multiplicity.