Realization of big centralizers of minimal aperiodic actions on the Cantor set
María Isabel Cortéz, Samuel Petite · Discrete and Continuous Dynamical Systems · 2020
In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal \begin{document}$ \mathbb Z $\end{document} action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group \begin{document}$ G $\end{document} , the centralizer of any minimal aperiodic \begin{document}$ G $\end{document} -action on the Cantor set is a subgroup of the centralizer of a minimal \begin{document}$ \mathbb Z $\end{document} -action.