Spectral theory of spin substitutions

Abstract

<p style='text-indent:20px;'>We introduce substitutions in <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{Z}}^m $\end{document}</tex-math></inline-formula> which have non-rectangular domains based on an endomorphism <inline-formula><tex-math id="M2">\begin{document}$ Q $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{Z}}^m $\end{document}</tex-math></inline-formula> and a set <inline-formula><tex-math id="M4">\begin{document}$ {\mathcal D} $\end{document}</tex-math></inline-formula> of coset representatives of <inline-formula><tex-math id="M5">\begin{document}$ {\mathbb{Z}}^m/Q{\mathbb{Z}}^m $\end{document}</tex-math></inline-formula>, which we call digit substitutions. Using a finite abelian 'spin' group we define 'spin digit substitutions' and their subshifts <inline-formula><tex-math id="M6">\begin{document}$ ({\Sigma}, {\mathbb{Z}}^m) $\end{document}</tex-math></inline-formula>. Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an <inline-formula><tex-math id="M7">\begin{document}$ m $\end{document}</tex-math></inline-formula>-dimensional odometer are given, inducing a complete decomposition of the function space <inline-formula><tex-math id="M8">\begin{document}$ L^{2}({\Sigma},\mu) $\end{document}</tex-math></inline-formula>. This enables the use of group characters in <inline-formula><tex-math id="M9">\begin{document}$ {\widehat{G}} $\end{document}</tex-math></inline-formula> 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.</p>