The Ellis semigroup of bijective 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.

In this version we improve substantially the description of the full Ellis semigroup $E(X)$. For primitive aperiodic bijective substitutions whose classical height coincides with its generalised height we show that $E(X)$ isomorphic to a semi-direct product of $E^{fib}(X)$ by $Y$