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Invariant measures for reducible generalized Bratteli diagrams

Sergey Bezuglyi, Olena Karpel, Jan Kwiatkowski

Abstract

In 2010, Bezuglyi, Kwiatkowski, Medynets and Solomyak [Ergodic Theory Dynam. Systems 30 (2010), no.4, 973-1007] found a complete description of the set of probability ergodic tail invariant measures on the path space of a standard (classical) stationary reducible Bratteli diagram. It was shown that every distinguished eigenvalue for the incidence matrix determines a probability ergodic invariant measure. In this paper, we show that this result does not hold for stationary reducible generalized Bratteli diagrams. We consider classes of stationary and non-stationary reducible generalized Bratteli diagrams with infinitely many simple standard subdiagrams, in particular, with infinitely many odometers as subdiagrams. We characterize the sets of all probability ergodic invariant measures for such diagrams and study partial orders under which the diagrams can support a Vershik homeomorphism.

Invariant measures for reducible generalized Bratteli diagrams

Abstract

In 2010, Bezuglyi, Kwiatkowski, Medynets and Solomyak [Ergodic Theory Dynam. Systems 30 (2010), no.4, 973-1007] found a complete description of the set of probability ergodic tail invariant measures on the path space of a standard (classical) stationary reducible Bratteli diagram. It was shown that every distinguished eigenvalue for the incidence matrix determines a probability ergodic invariant measure. In this paper, we show that this result does not hold for stationary reducible generalized Bratteli diagrams. We consider classes of stationary and non-stationary reducible generalized Bratteli diagrams with infinitely many simple standard subdiagrams, in particular, with infinitely many odometers as subdiagrams. We characterize the sets of all probability ergodic invariant measures for such diagrams and study partial orders under which the diagrams can support a Vershik homeomorphism.
Paper Structure (10 sections, 9 theorems, 43 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions on Bratteli diagrams
  4. Subdiagrams and measure extension
  5. Invariant measures on stationary standard Bratteli diagrams
  6. Invariant measures on stationary irreducible generalized Bratteli diagrams
  7. Measure extension for generalized Bratteli diagrams
  8. Reducible Bratteli diagrams with infinitely many odometers
  9. Some classes of generalized Bratteli diagrams with infinitely many odometers
  10. Stationary generalized Bratteli diagrams with infinitely many odometers.

Key Result

Theorem 2.7

Let $B = (V,E)$ be a Bratteli diagram (generalized or standard) with the sequence of incidence matrices $(F_n)$. Then:

Figures (1)

  • Figure :

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 21 more