Equipped graded graphs, projective limits of simplices, and their boundaries

TL;DR

This paper develops a unified theory of equipped graded graphs and projective limits of simplices, addressing the problem of describing probability measures and boundaries relevant to Markov chains, AF-algebras, and group characters.

Contribution

It introduces a new framework unifying boundaries in Markov processes, AF-algebras, and group theory through equipped graphs and projective limits.

Findings

Unified interpretation of Martin, Choquet, and Dynkin boundaries

Introduction of the concept of standardness in projective limits

Analysis of lacunarization in equipped Bratteli diagrams

Abstract

In this paper, we develop a theory of equipped graded graphs (or Bratteli diagrams) and an alternative theory of projective limits of finite-dimensional simplices. An equipment is an additional structure on the graph, namely, a system of "cotransition" probabilities on the set of its paths. The main problem is to describe all probability measures on the path space of a graph with given cotransition probabilities; it goes back to the problem, posed by E.~B.~Dynkin in the 1960s, of describing exit and entrance boundaries for Markov chains. The most important example is the problem of describing all central measures, to which one can reduce the problems of describing states on AF-algebras or characters on locally finite groups. We suggest an unification of the whole theory, an interpretation of the notions of Martin, Choquet, and Dynkin boundaries in terms of equipped graded graphs and in terms of the theory of projective limits of simplices. In the last section, we study the new notion of "standardness" of projective limits of simplices and of equipped Bratteli diagrams, as well as the notion of "lacunarization."