Decisive Bratteli-Vershik models

TL;DR

This paper characterizes when zero-dimensional systems have decisive Bratteli-Vershik models, showing that such models exist precisely when aperiodic points are dense or nearly dense, providing a clear criterion for model decisiveness.

Contribution

It establishes a necessary and sufficient condition for the existence of decisive Bratteli-Vershik models in zero-dimensional systems.

Findings

Decisive models exist if aperiodic points are dense or nearly dense.

Provides a complete characterization of decisiveness in Bratteli-Vershik models.

Connects the structure of aperiodic points to the uniqueness of the Vershik map extension.

Abstract

In this paper we focus on Bratteli-Vershik models of general compact zero-dimensional systems with the action of a homeomorphism. An ordered Bratteli diagram is called decisive if the corresponding Vershik map prolongs in a unique way to a homeomorphism of the whole path space of the Bratteli diagram. We prove that a compact invertible zero-dimensional system has a decisive Bratteli-Vershik model if and only if the set of aperiodic points is either dense, or its closure misses one periodic orbit.