Inverse limit method for generalized Bratteli diagrams and invariant measures

TL;DR

This paper develops an inverse limit approach to describe all ergodic tail invariant measures on generalized Bratteli diagrams, applying it to specific classes like the infinite Pascal graph and odometers, revealing detailed measure structures.

Contribution

It introduces a general inverse limit method for characterizing ergodic measures on generalized Bratteli diagrams and applies it to explicit classes, including the infinite Pascal graph.

Findings

Explicit description of ergodic measures for the infinite Pascal graph

Formulas for measure values on cylinder sets

Analysis of measure extension and Vershik map properties

Abstract

Generalized Bratteli diagrams with a countable set of vertices in every level are models for aperiodic Borel automorphisms. This paper is devoted to the description of all ergodic probability tail invariant measures on the path spaces of generalized Bratteli diagrams. Such measures can be identified with inverse limits of infinite-dimensional simplices associated with levels in generalized Bratteli diagrams. Though this method is general, we apply it to several classes of reducible generalized Bratteli diagrams. In particular, we explicitly describe all ergodic tail invariant probability measures for (i) the infinite Pascal graph and give the formulas for the values of such measures on cylinder sets, (ii) generalized Bratteli diagrams formed by a countable set of odometers, (iii) reducible generalized Bratteli diagrams with uncountable set of ergodic tail invariant probability measures. We also consider the method of measure extension by tail invariance from subdiagrams. We discuss the properties of the Vershik map defined on reducible generalized Bratteli diagrams.