A classification of nonexpansive Bratteli-Vershik systems

TL;DR

This paper classifies nonexpansive Bratteli-Vershik systems into distinct types, correcting previous definitions and establishing their properties, relations, and examples, including the disjointness of conjugates to well timed and untimed systems.

Contribution

It redefines key classes of nonexpansive Bratteli-Vershik systems, clarifies their relationships, and proves the disjointness of systems conjugate to well timed and untimed systems.

Findings

Class of nonexpansive BV systems is the union of conjugates to well timed and untimed systems.

Several examples illustrate the classification and properties.

Corrects previous definitions and introduces new classes based on timing and indistinguishability.

Abstract

We study simple, properly ordered nonexpansive Bratteli-Vershik ($BV$) systems. Correcting a mistake in an earlier paper, we redefine the classes standard nonexpansive ($SNE$) and strong standard nonexpansive ($SSNE$). We define also the classes of very well timed and well timed systems, their opposing classes of untimed and very untimed systems (which feature, as subclasses of "Case (2)", in the work of Downarowicz and Maass as well as Hoynes on expansiveness of $BV$ systems of finite topological rank), and several related classes according to the existence of indistinguishable pairs (of some "depth") and their synchronization ("common cuts"). We establish some properties of these types of systems and some relations among them. We provide several relevant examples, including a problematic one that is conjugate to a well timed system while also (vacuously) in the classes "Case (2)". We prove that the class of all simple, properly ordered nonexpansive $BV$ systems is the disjoint union of the ones conjugate to well timed systems and those conjugate to untimed systems, thereby showing that nonexpansiveness in $BV$ systems arises in one of two mutually exclusive ways.