The problem of describing central measures on path space of graded graphs

TL;DR

This paper introduces a new method using the internal metric to describe invariant measures on path spaces of graded graphs, aiding in characterizing group characters and AF-algebra traces.

Contribution

It develops a novel approach based on the internal metric and standard filtrations to effectively enumerate ergodic invariant measures on graph path spaces.

Findings

The internal metric guarantees relative compactness, enabling constructive enumeration of ergodic measures.

Application of the method recovers classical theorems on invariant measures.

The approach links filtrations properties to measure classification in graph path spaces.

Abstract

We suggest a new method of describing invariant measures on Markov compacta and path spaces of graphs, and thus of describing characters of some groups and traces of AF-algebras. The method relies on properties of filtrations associated with the graph and, in particular, on the notion of a standard filtration. The main tool is the so-called internal metric that we introduce on simplices of measures; it is an iterated Kantorovich metric, and the key result is that the relative compactness in this metric guarantees a constructive enumeration of the ergodic invariant measures. Applications include a number of classical theorems on invariant measures