# Automorphisms of automatic shifts

## TL;DR

This paper investigates the automorphism groups of constant length substitution shifts, revealing that roots of identity are letter exchanges and automorphisms are related to twisted compressions, with implications for their topological factors.

## Contribution

It characterizes the automorphism groups of constant length substitution shifts, including roots of identity and their topological factors, advancing understanding of their structure.

## Key findings

- Roots of identity are letter exchanging maps.
- Nontrivial automorphisms are twisted compressions.
- Topological factors are conjugate to substitution shifts.

## Abstract

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from {\em twisted} compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a letter-to-letter code.