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Subdiagrams and invariant measures for generalized Bratteli diagrams

Sergey Bezuglyi, Palle Jorgensen, Olena Karpel, Thiago Raszeja, Shrey Sanadhya

TL;DR

This work develops a comprehensive framework for tail invariant measures on path spaces of generalized Bratteli diagrams with countably infinite vertex sets, focusing on how to extend measures from subdiagrams to the ambient diagram and when such extensions remain finite. It provides necessary and sufficient conditions for finiteness of extensions via incidence matrices and associated row-stochastic matrices, and offers constructive, step-by-step procedures for extension and for approximating invariant measures through subdiagrams. The authors analyze several diagram classes (simple, stationary, and fat odometer) and reveal phenomena absent in standard finite-vertex Bratteli diagrams, including examples with no probability tail invariant measures and intricate convergence behavior under subdiagram approximations. A key part of the contribution is the systematic treatment of path-space structure, the 0-1 procedure relating edge and vertex subdiagrams, and convergence notions for measures, which together connect combinatorial diagram data with measure-theoretic dynamics in Borel/Cantor settings.

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.

Subdiagrams and invariant measures for generalized Bratteli diagrams

TL;DR

This work develops a comprehensive framework for tail invariant measures on path spaces of generalized Bratteli diagrams with countably infinite vertex sets, focusing on how to extend measures from subdiagrams to the ambient diagram and when such extensions remain finite. It provides necessary and sufficient conditions for finiteness of extensions via incidence matrices and associated row-stochastic matrices, and offers constructive, step-by-step procedures for extension and for approximating invariant measures through subdiagrams. The authors analyze several diagram classes (simple, stationary, and fat odometer) and reveal phenomena absent in standard finite-vertex Bratteli diagrams, including examples with no probability tail invariant measures and intricate convergence behavior under subdiagram approximations. A key part of the contribution is the systematic treatment of path-space structure, the 0-1 procedure relating edge and vertex subdiagrams, and convergence notions for measures, which together connect combinatorial diagram data with measure-theoretic dynamics in Borel/Cantor settings.

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.
Paper Structure (30 sections, 47 theorems, 294 equations, 5 figures)

This paper contains 30 sections, 47 theorems, 294 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Why do we need Bratteli diagrams?
  3. What do we study in the paper?
  4. What is a Bratteli diagram?
  5. Motivation
  6. Three methods for constructing tail invariant measures
  7. The main results and outline of the paper
  8. Basics on Bratteli diagrams and tail invariant measures
  9. Subdiagrams and measure extension from a subdiagram
  10. Vertex and edge subdiagrams and measure extension
  11. Measure extension from a subdiagram.
  12. Measure extension from a vertex subdiagram
  13. Measure extension from an edge subdiagram
  14. Measure extension for certain classes of generalized Bratteli diagrams
  15. Measure extension from simple subdiagrams
  16. ...and 15 more sections

Key Result

Theorem 2.7

Let $B = (V,E)$ be a Bratteli diagram (generalized or standard) with the sequence of incidence matrices $(F_n)_{n \in \mathbb N_0}$.

Figures (5)

  • Figure 1: The diagram $B(2^n)$
  • Figure 2: $Y = X_B$.
  • Figure 3: $cl(\Omega) = cl(Y) = X_B$.
  • Figure 4: Diagrams $B$ (on the left) and $\tilde{B}$ (on the right) from the \ref{['ex:0-1proc']}
  • Figure 5: The Bratteli diagram for \ref{['example_measure_convergence_BD_inf']}.

Theorems & Definitions (147)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8: Generalized Perron-Frobenius theorem
  • Theorem 2.9
  • Remark 2.10
  • ...and 137 more