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Measure transfer and $S$-adic developments for subshifts

Nicolas Bédaride, Arnaud Hilion, Martin Lustig

TL;DR

The paper develops a comprehensive $S$-adic framework to study subshifts by tracking measure transfer along non-erasing morphisms via the measure cone and the letter-frequency cone. It shows that for any non-erasing morphism, the measure transfer $oldsymbol{ ext M}^{ ext M}$ maps $oldsymbol{ ext M}(X)$ onto $oldsymbol{ ext M}(\sigma(X))$, and that recognizability yields injectivity; for totally recognizable, everywhere-growing directive sequences the map from vector towers to measures is a linear bijection, enabling a precise description of invariant measures through letter frequencies. These tools are then applied to construct large classes of dynamical systems: (i) minimal subshifts with zero entropy and infinitely many ergodic measures, and (ii) a side result of a minimal, aperiodic, uniquely ergodic subshift with all level alphabets of size $d$ where the bottom morphisms are non-recognizable. The authors also analyze non-recognizable directive sequences via geometric quotient constructions inspired by Thurston theory, showing how non-injectivity can arise naturally in $S$-adic developments and yield rich ergodic structures. Overall, the framework provides a direct, algebraic route to linking combinatorial data from directive sequences with ergodic and topological properties of subshifts, including entropy, orbit structure, and the spectrum of invariant measures.

Abstract

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.

Measure transfer and $S$-adic developments for subshifts

TL;DR

The paper develops a comprehensive -adic framework to study subshifts by tracking measure transfer along non-erasing morphisms via the measure cone and the letter-frequency cone. It shows that for any non-erasing morphism, the measure transfer maps onto , and that recognizability yields injectivity; for totally recognizable, everywhere-growing directive sequences the map from vector towers to measures is a linear bijection, enabling a precise description of invariant measures through letter frequencies. These tools are then applied to construct large classes of dynamical systems: (i) minimal subshifts with zero entropy and infinitely many ergodic measures, and (ii) a side result of a minimal, aperiodic, uniquely ergodic subshift with all level alphabets of size where the bottom morphisms are non-recognizable. The authors also analyze non-recognizable directive sequences via geometric quotient constructions inspired by Thurston theory, showing how non-injectivity can arise naturally in -adic developments and yield rich ergodic structures. Overall, the framework provides a direct, algebraic route to linking combinatorial data from directive sequences with ergodic and topological properties of subshifts, including entropy, orbit structure, and the spectrum of invariant measures.

Abstract

Based on previous work of the authors, to any -adic development of a subshift a "directive sequence" of commutative diagrams is associated, which consists at every level of the measure cone and the letter frequency cone of the level subshift associated canonically to the given -adic development. The issuing rich picture enables one to deduce results about with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer , an -adic development of a minimal, aperiodic, uniquely ergodic subshift , where all level alphabets have cardinality , while none of the bottom level morphisms is recognizable in its level subshift .
Paper Structure (17 sections, 29 theorems, 82 equations)

This paper contains 17 sections, 29 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Terminology, notation, conventions and some quotes
  3. Standard terminology from symbolic dynamics
  4. Measures on subshifts via vector towers on directive sequences
  5. The measure transfer and its injectivity for recognizable morphisms
  6. The measure transfer and some results from PartI
  7. Recognizable morphisms and some related properties
  8. Injectivity of the measure transfer for recognizable morphisms
  9. The measure transfer via vector towers
  10. Measure towers and vector towers
  11. Directive sequences with "small" intermediate letter frequency cones
  12. Minimal subshifts with zero entropy and infinitely many ergodic probability measures
  13. Non-recognizable directive sequences
  14. The basic geometric quotient construction
  15. The "inverse" geometric quotient construction
  16. ...and 2 more sections

Key Result

Proposition 1.1

For any everywhere growing directive sequence $\overleftarrow \sigma$ there is a canonical linear bijection between the cone $\mathcal{V}(\overleftarrow \sigma)$ of vector towers and the cone $\mathcal{M}(\overleftarrow \sigma)$ of measure towers on $\overleftarrow \sigma$, given by the letter frequ with $\vec{v}_n = \vec{v}(\mu_n) =(\mu_n([a_k]))_{a_k \in \mathcal{A}_n}$ for all levels $n \geq 0$

Theorems & Definitions (59)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 1.7
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.4
  • ...and 49 more