Non-homeomorphic topological rank and expansiveness

TL;DR

This paper extends the concept of topological rank to all Cantor minimal continuous surjections and proves that those with finite rank have expansive natural extensions, broadening understanding of dynamical systems.

Contribution

It introduces a topological rank for Cantor minimal continuous surjections and demonstrates that finite rank implies expansiveness of their natural extensions.

Findings

Finite topological rank implies expansive natural extension.

Extension of rank concept from homeomorphisms to surjections.

Broader class of systems shown to have expansive behavior.

Abstract

Downarowicz and Maass (2008) have shown that every Cantor minimal homeomorphism with finite topological rank $K > 1$ is expansive. Bezuglyi, Kwiatkowski and Medynets (2009) extended the result to non-minimal cases. On the other hand, Gambaudo and Martens (2006) had expressed all Cantor minimal continuou surjections as the inverse limit of graph coverings. In this paper, we define a topological rank for every Cantor minimal continuous surjection, and show that every Cantor minimal continuous surjection of finite topological rank has the natural extension that is expansive.