Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tower power for $S$-adics

Nicolas Bédaride, Arnaud Hilion, Martin Lustig

Abstract

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and $S$-adic systems in symbolic dynamics. We then produce as rather direct application an $S$-adic system (with finite set of substitutions $S$ on $d$ letters) that is minimal and has $d$ distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $μ([w])$, for any word $w \in \cal A^*$ and any invariant measure $μ$ on the subshift $X_σ$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $σ: \cal A^* \to \cal A^*$. Several examples are considered in detail, and model computations are presented.

Tower power for $S$-adics

Abstract

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and -adic systems in symbolic dynamics. We then produce as rather direct application an -adic system (with finite set of substitutions on letters) that is minimal and has distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure , for any word and any invariant measure on the subshift defined by any everywhere growing but not necessarily primitive or irreducible substitution . Several examples are considered in detail, and model computations are presented.

Paper Structure

This paper contains 18 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subshifts
  4. Substitutions
  5. $S$-adic sequences
  6. Results from BHL1
  7. The general setting
  8. Application to substitutions
  9. Cylinder measures
  10. Minimal subshifts with many ergodic measures
  11. The general construction procedure
  12. Reduction to finite $S$
  13. Some questions
  14. Determination of cylinder weights for substitutions
  15. Examples
  16. ...and 3 more sections

Key Result

Proposition 1.1

(1) For any integer $d \geq 1$ there exists a directive sequence $\sigma = \sigma_0 \circ \sigma_1 \circ \ldots$, with level alphabets $\mathcal{A}_n$ all of cardinality $d$, such that the associated subshift $X_\sigma$ is minimal and supports $d$ distinct invariant ergodic probability measures. (2)

Theorems & Definitions (29)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3: BD
  • Lemma 2.4
  • Definition 3.1
  • Theorem 3.2: BHL1
  • Lemma 3.3
  • Proposition 3.4
  • ...and 19 more