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Refinement of Bratteli-Vershik models

Takashi Shimomura

Abstract

In the zero-dimensional systems, the Bratteli-Vershik models can be built upon certain closed sets that are called `quasi-sections' in this article. There exists a bijective correspondence between the topological conjugacy classes of triples of zero-dimensional systems and quasi-sections and the topological conjugacy classes of Bratteli-Vershik models. Therefore, we can get refined Bratteli-Vershik models if we get certain refined quasi-sections. The basic sets are such refined quasi-sections that bring `closing property' on the corresponding Bratteli-Vershik models. We show a direct proof on the existence of basic sets. Thorough investigations on quasi-sections and basic sets are done. Furthermore, it would be convenient for the Bratteli-Vershik models to concern minimal sets. To this point, we show the existence of the Bratteli-Vershik models whose minimal sets are properly ordered. On the other hand, we can get certain refinements with respect to the Bratteli-Vershikizability condition or the decisiveness.

Refinement of Bratteli-Vershik models

Abstract

In the zero-dimensional systems, the Bratteli-Vershik models can be built upon certain closed sets that are called `quasi-sections' in this article. There exists a bijective correspondence between the topological conjugacy classes of triples of zero-dimensional systems and quasi-sections and the topological conjugacy classes of Bratteli-Vershik models. Therefore, we can get refined Bratteli-Vershik models if we get certain refined quasi-sections. The basic sets are such refined quasi-sections that bring `closing property' on the corresponding Bratteli-Vershik models. We show a direct proof on the existence of basic sets. Thorough investigations on quasi-sections and basic sets are done. Furthermore, it would be convenient for the Bratteli-Vershik models to concern minimal sets. To this point, we show the existence of the Bratteli-Vershik models whose minimal sets are properly ordered. On the other hand, we can get certain refinements with respect to the Bratteli-Vershikizability condition or the decisiveness.

Paper Structure

This paper contains 13 sections, 29 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Basics of quasi-sections and basic sets
  4. Basic sets
  5. Quasi-simple basic set
  6. Quasi-section and K--R refinement
  7. Bratteli--Vershik models and main basic results
  8. Bratteli--Vershik models
  9. A link with K--R refinements
  10. Subsystems and Bratteli--Vershik models
  11. The proofs
  12. Decisiveness, quasi-sections, and basic sets
  13. Applications

Key Result

Theorem 1.1

There exists a conjugacy-preserving bijective correspondence between topological conjugacy classes of triples of zero-dimensional systems and quasi-sections and topological conjugacy classes of Bratteli--Vershik models.

Theorems & Definitions (69)

  • Theorem 1.1: Putnam Putnam89TheCstarAlgebrasAssociatedWithMinimalHomeomorphismsOfTheCantorSet
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 59 more