A classification of nonexpansive Bratteli-Vershik systems

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.