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Substitution-dynamics and invariant measures for infinite alphabet-path space

Sergey Bezuglyi, Palle E. T. Jorgensen, Shrey Sanadhya

Abstract

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.

Substitution-dynamics and invariant measures for infinite alphabet-path space

Abstract

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.
Paper Structure (18 sections, 24 theorems, 93 equations, 3 figures)

This paper contains 18 sections, 24 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries
  3. Substitutions on infinite alphabet
  4. Borel dynamical system
  5. Left determined substitutions
  6. Kakutani-Rokhlin towers for infinite substitutions.
  7. Substitutions on infinite alphabet as Borel dynamical systems
  8. Nested sequence of complete sections
  9. Generalized Bratteli-Vershik models for infinite substitutions.
  10. Generalized Bratteli diagrams
  11. Stationary generalized Bratteli-Vershik model for substitutions on infinite alphabet
  12. Invariant measure for infinite substitution via Bratteli diagrams
  13. Tail invariant measure for stationary Bratteli diagrams
  14. From tail invariant measure to shift-invariant measure
  15. Examples
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\sigma$ be a bounded size left determined substitution on a countably infinite alphabet and $(X_\sigma, T)$ be the corresponding subshift. Then there exists a stationary ordered generalized Bratteli diagram $B = (V, E, \geq)$ and a Vershik map $\varphi : Y_B \rightarrow Y_B$ such that $(X_\sigm

Figures (3)

  • Figure 1: Example of a Bratteli diagram: levels, vertices, and edges (see Definition \ref{['def GBD']})
  • Figure 2: Generalized B-V model for Example 8.1.
  • Figure 3: Generalized B-V model of the Random walk on ${\mathbb Z}$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Feldman--Moore FeldmanMoore_1977
  • Definition 2.7
  • Definition 3.1
  • ...and 54 more