# The Ellis semigroup of bijective substitutions

## TL;DR

This paper investigates the structure of the Ellis semigroup for dynamical systems from primitive aperiodic bijective substitutions, revealing that their virtual automorphism group coincides with the automorphism group.

## Contribution

It introduces methods to compute the fiber-preserving part of the Ellis semigroup for these systems and characterizes their automorphism groups.

## Key findings

- Computed the Ellis semigroup for substitution shifts
- Showed the virtual automorphism group equals the automorphism group
- Provided new tools for analyzing dynamical systems from substitutions

## Abstract

For topological dynamical systems $(X,T,\sigma)$ with abelian group $T$ which admit an equicontinuous factor $\pi:(X,T,\sigma)\to (Y,T,\delta)$ the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $E^{fib}(X)$ of elements which preserve the fibres of $\pi$. We establish methods to compute $E^{fib}(X)$ and use them to determine the Ellis semigroup of dynamical systems arising from primitive aperiodic bijective substitutions. As an application we show that for these substitution shifts, the virtual automorphism group is isomorphic to the classical automorphism group.