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The measure transfer for subshifts induced by a morphism of free monoids

Nicolas Bédaride, Arnaud Hilion, Martin Lustig

TL;DR

This work develops a comprehensive framework for transferring invariant measures along non-erasing morphisms between subshifts. By factorizing a morphism into subdivision and letter-to-letter parts, the authors define the measure transfer map $\sigma^{\mathcal{M}}$ and establish its linearity, continuity, and functoriality, with surjectivity on image measures and ergodicity preservation. They provide explicit formulas to compute transferred cylinder measures, notably via essential occurrences, and prove a key injectivity result: if the morphism is injective on the shift-orbits of the preimage subshift, then the induced measure transfer is injective. The paper also clarifies the relationships among injectivity, shift-period preservation, and recognizability, and treats the special case of letter-to-letter morphisms where the transfer reduces to the push-forward, highlighting implications for S-adic expansions and related constructions in symbolic dynamics.

Abstract

Every non-erasing monoid morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $σ_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(σ(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(σ(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $σ(X) \subset \mathcal{B}^{\mathbb{Z}}$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure $σ_X^{\mathcal{M}(μ)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $σ^{\mathcal{M}_X}$ is injective. \smallskip The assumption on $σ$ that it is ``injective on the shift-orbits of $X$'' is strictly weaker than ``recognizable in $X$'', and strictly stronger than ``recognizable for aperiodic points in $X$''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map $σ_X^{\mathcal{M}}$.

The measure transfer for subshifts induced by a morphism of free monoids

TL;DR

This work develops a comprehensive framework for transferring invariant measures along non-erasing morphisms between subshifts. By factorizing a morphism into subdivision and letter-to-letter parts, the authors define the measure transfer map and establish its linearity, continuity, and functoriality, with surjectivity on image measures and ergodicity preservation. They provide explicit formulas to compute transferred cylinder measures, notably via essential occurrences, and prove a key injectivity result: if the morphism is injective on the shift-orbits of the preimage subshift, then the induced measure transfer is injective. The paper also clarifies the relationships among injectivity, shift-period preservation, and recognizability, and treats the special case of letter-to-letter morphisms where the transfer reduces to the push-forward, highlighting implications for S-adic expansions and related constructions in symbolic dynamics.

Abstract

Every non-erasing monoid morphism induces a {\em measure transfer map} between the measure cones and , associated to any subshift and its image subshift respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure on any cylinder (for ) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism is injective on the shift-orbits of some subshift , then is injective. \smallskip The assumption on that it is ``injective on the shift-orbits of '' is strictly weaker than ``recognizable in '', and strictly stronger than ``recognizable for aperiodic points in ''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map .
Paper Structure (17 sections, 25 theorems, 93 equations, 1 figure)

This paper contains 17 sections, 25 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Standard terminology and well known facts
  4. "Not so standard" basic facts and terminology
  5. About injectivity
  6. Typical injectivity problems
  7. Shift-period preservation
  8. The measure transfer
  9. Subdivision morphisms
  10. Letter-to-letter morphisms
  11. The induced measure morphisms
  12. Basic properties of the measure transfer map
  13. Evaluation of the transferred measure $\sigma^\mathcal{M}(\mu)$
  14. A first example for the measure transfer
  15. An alternative evaluation method
  16. ...and 2 more sections

Key Result

Proposition 1.1

Let $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ be any non-erasing morphism, and let $X \subseteq \mathcal{A}^\mathbb Z$ be any subshift over $\mathcal{A}$, with image subshift $Y := \sigma(X) \subseteq \mathcal{B}^\mathbb Z$. Then the induced measure transfer map $\sigma^\mathcal{M}$ restricts/co-res which has the following properties: (In addition, the map $\sigma_X^\mathcal{M}$ is surjective, se

Figures (1)

  • Figure 1:

Theorems & Definitions (70)

  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Example 1.6
  • Remark 2.1
  • Example 2.3
  • Lemma 2.4
  • proof
  • ...and 60 more