Spectral theory of spin substitutions

Abstract

We introduce qubit substitutions in $\mathbb{Z}^m$, which have non-rectangular domains based on an endomorphism $Q$ of $\mathbb{Z}^m$ and a set $\mathcal{D}$ of coset representatives of $\mathbb{Z}^m/Q\mathbb{Z}^m$. We then focus on a specific family of qubit substitutions which we call spin substitutions, whose combinatorial definition requires a finite abelian group $G$ as its spin group. We investigate the spectral theory of the underlying subshift $(\Sigma,\mathbb{Z}^m)$. Under certain assumptions, we show that it is measure-theoretically isomorphic to a group extension of an $m$-dimensional odometer, which induces a complete decomposition of the function space $L^{2}(\Sigma,\mu)$ . This enables one to use group characters in $\widehat{G}$ to derive substitutive factors and carry out a spectral analysis on 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.