Asymptotic theory of path spaces of graded graphs and its applications

TL;DR

This survey explores the asymptotic structure of path spaces in graded graphs, introducing a new notion of standardness that classifies filtrations and impacts the parametrization of invariant measures.

Contribution

It introduces a new concept of standardness for filtrations and provides a general criterion for standardness, enhancing understanding of asymptotic behaviors in graded graphs.

Findings

Standard and non-standard filtrations are distinguished by their asymptotic behavior.

A criterion for determining standardness in filtrations is established.

Applications demonstrate the broad relevance of the standardness concept.

Abstract

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The main subject is the asymptotic structure of filtrations and a new notion of standardness. All graded graphs and all filtrations of Borel or measure spaces can be divided into two classes: the standard ones, which have a regular behavior at infinity, and the other ones. Depending on this property, the list of invariant measures can either be well parameterized or have no good parametrization at all. One of the main results is a general standardness criterion for filtrations. We consider some old and new examples which illustrate the usefulness of this point of view and the breadth of its applications.