Almost automorphic and bijective factors of substitution shifts

TL;DR

This paper provides a complete algebraic characterization of constant length substitution shifts that have almost automorphic or bijective factors, using Green's relations in a finite semigroup framework.

Contribution

It introduces a novel algebraic approach to identify almost automorphic and bijective factors in substitution shifts via Green's R- and L-relations.

Findings

Characterization of almost automorphic factors using Green's R-relation

Characterization of bijective factors using Green's L-relation

Results are constructive and applicable to substitution shifts

Abstract

In this article we completely characterise constant length substitution shifts which have an almost automorphic factor, or which have a bijective substitution factor. Our approach is algebraic: we study these dynamical properties in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green's R-relation, and the existence of bijective factors in terms of Green's L-relation. Our results are constructive.