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

TL;DR

This paper characterizes when a Cantor dynamical system can be embedded in the real line with zero derivative everywhere, revealing a large class of such systems and providing a constructive approach.

Contribution

It provides a complete characterization and construction method for Cantor systems embeddable in  with vanishing derivative, solving a longstanding problem.

Findings

A complete criterion for embedding in  with zero derivative

Construction of a refining sequence of partitions

Identification of a large class of such systems

Abstract

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