Finite rank Bratteli--Vershik homeomorphisms are expansive

TL;DR

This paper extends the understanding of zero-dimensional systems by showing that finite-rank systems are either expansive or conjugate to infinite odometer systems, building on prior results for minimal and aperiodic cases.

Contribution

It generalizes previous results by demonstrating that all finite-rank zero-dimensional systems are either expansive or odometers, broadening the scope of known dynamical behaviors.

Findings

Finite-rank zero-dimensional systems are either expansive or odometers.

Extends previous results from minimal and aperiodic systems to all zero-dimensional systems.

Uses similar methods to earlier proofs for broader classes of systems.

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 aperiodic cases. In this paper, we show that all finite-rank zero-dimensional systems are expansive or have infinite odometer systems; this is an extension of the two aforementioned results. Nevertheless, the methods follow similar approaches.