On Topological Rank of Factors of Cantor Minimal Systems

TL;DR

This paper proves that if a minimal dynamical system on a Cantor set has finite topological rank, then all its minimal Cantor factors also have finite topological rank, answering a previously open question.

Contribution

It establishes that finite topological rank is preserved under taking minimal Cantor factors in such systems.

Findings

Finite topological rank implies all factors have finite rank.

Answers an open question about rank preservation.

Provides a structural insight into Cantor minimal systems.

Abstract

A Cantor minimal system is of finite topological rank if it has a Bratteli-Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite.