Spectral theory of spin substitutions

TL;DR

This paper develops a spectral theory framework for spin substitutions in $ Z^m$, revealing how their spectral measures can be classified and analyzed using group characters and measure-theoretic isomorphisms.

Contribution

It introduces spin substitutions with a novel combinatorial structure and provides criteria for their spectral types, advancing understanding of their spectral properties.

Findings

Spectral measures can be classified into pure point, absolutely continuous, and singular continuous types.

The subshift is measure-theoretically isomorphic to a group extension of an odometer.

Criteria for the existence of different spectral types are established.

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.