# Almost automorphic and bijective factors of substitution shifts

## TL;DR

This paper characterizes substitution shifts with specific dynamical properties using algebraic methods involving Green’s relations.

## Contribution

The paper provides a complete algebraic characterization of almost automorphic and bijective substitution factors using Green’s relations.

## Key findings

- Almost automorphic factors are characterized using Green’s R-relation.
- Bijective factors are characterized using Green’s L-relation.
- The results are constructive and apply to constant length substitution shifts.

## Abstract

In this article we completely characterise constant length substitution shifts which have a proper almost automorphic factor, or which have a bijective substitution factor such that the factor map is injective on at least one point. Our approach is algebraic: we characterise 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 \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\mathcal {R}}$$\end{document}R-relation and the existence of bijective factors in terms of Green’s \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${{\mathcal {L}}}$$\end{document}L-relation. Our results are constructive.