Spectral theory of spin substitutions

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