Subdiagrams and invariant measures for generalized Bratteli diagrams

TL;DR

This paper investigates invariant measures on generalized Bratteli diagrams with countably infinite vertices, establishing conditions for measure extension, analyzing specific classes, and providing examples where no probability measures exist.

Contribution

It introduces criteria for extending tail invariant measures from subdiagrams to the entire diagram in the generalized setting, including new phenomena absent in finite cases.

Findings

Necessary and sufficient conditions for measure extension.

Explicit examples of diagrams with no probability tail invariant measures.

Procedures for approximating invariant measures via subdiagrams.

Abstract

The results of this paper contribute to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior.