A Cantor dynamical system is slow if and only if all its finite orbits are attracting

Abstract

In this paper we completely solve the problem of when a Cantor dynamical system \begin{document}$ (X, f) $\end{document} can be embedded in \begin{document}$ \mathbb{R} $\end{document} with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of \begin{document}$ X $\end{document} which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems.