Finite-rank Bratteli-Vershik diagrams are expansive -- a new proof

TL;DR

This paper provides a clearer, more accessible proof that finite-rank Bratteli-Vershik diagrams produce systems that are either odometers or expansive, and improves the bounds related to this classification.

Contribution

It offers a new, more transparent proof of a key classification theorem and improves the bounds on system types for finite-rank Bratteli-Vershik diagrams.

Findings

New proof is more transparent and easier to understand.

Improved lower bound on the classification of systems.

Conjecture that the new bound is optimal.

Abstract

In the paper "Finite-rank Bratteli-Vershik diagrams are expansive" [DM], Downarowicz and Maass proved that the Cantor minimal system associated to a properly ordered Bratteli diagram of finite rank is either an odometer system or an expansive system. We give a new proof of this truly remarkable result which we think is more transparent and easier to understand. We also address the question (QUESTION 1) raised in [DM] and we find a better (i.e. lower) bound than the one given in [DM]. In fact, we conjecture that the bound we have found is optimal.