[Paper Review] Invariant measures on finite rank subshifts

Nicolas Bédaride, Arnaud Hilion+1 more|arXiv (Cornell University)|Jul 19, 2020

Mathematical Dynamics and Fractals5 references4 citations

TL;DR

This paper establishes that for any subshift of finite $S$-rank, invariant measures are completely determined by their values on the letter cylinders—i.e., the measures of individual letters in the alphabet. The result follows from the structure of $S$-adic developments and the invariance of the cone of letter measures across levels, with key implications for symbolic dynamics and measure-theoretic rigidity in subshifts.

ABSTRACT

In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

Motivation & Objective

• To establish that invariant measures on finite $S$-rank subshifts are determined by their values on letter cylinders.
• To clarify the role of $S$-adic developments and incidence matrices in characterizing invariant measures.
• To demonstrate that the cone of letter measures remains constant across levels in thin $S$-adic developments.
• To provide a concise proof of measure determination via structural properties of subshifts and morphisms.
• To highlight the broader implications of this result for symbolic dynamics and ergodic theory.

Proposed method

• The authors use $S$-adic developments, where a subshift $X$ is generated by a sequence of morphisms $\sigma_n: \mathcal{A}_{n+1}^* \to \mathcal{A}_n^*$, to analyze invariant measures.
• They define the cone $\mathcal{C}_n$ of letter measures for each level-$n$ subshift $X_n$, and study the behavior of the map $\zeta_n: \mathcal{M}(X) \to \mathcal{C}_n$.
• The key technical tool is the incidence matrix $M(\sigma_n)$, which encodes the number of times each letter appears in the image of a morphism.
• The proof relies on the fact that for an everywhere growing $S$-adic development, the map $M(\sigma_n)$ maps $\mathcal{C}_{n+1}$ surjectively onto $\mathcal{C}_n$, and if $\dim \mathcal{C}_n$ is constant, then $\zeta_n$ is bijective.
• The authors apply a result from prior work showing that if all $\mathcal{C}_n$ have the same dimension $c(X)$, then $\zeta_0$ is bijective, implying that measures are determined by letter cylinder values.
• They extend the result to general finite $S$-rank subshifts using a level $n_0$ beyond which the dimension stabilizes, and apply a functorial property of morphisms to lift the measure determination to the original subshift.

Experimental results

Research questions

• RQ1 Under what conditions is an invariant measure on a subshift uniquely determined by its values on letter cylinders?
• RQ2 How does the structure of an $S$-adic development influence the dimension of the cone of invariant measures?
• RQ3 What is the role of the incidence matrix in determining measure rigidity in subshifts?
• RQ4 When does the cone of letter measures remain constant across levels in an $S$-adic development?
• RQ5 Can measure determination be extended from intermediate subshifts to the full subshift via morphism-induced maps?

Key findings

• For any thin subshift of finite $S$-rank, two invariant measures are equal if and only if they assign the same measure to each letter cylinder $[a_k]$.
• The dimension $c_n$ of the cone of letter measures $\mathcal{C}_n$ stabilizes to $c(X)$ for all $n \geq n_0$, ensuring measure determination from a finite set of cylinders.
• If all incidence matrices $M(\sigma_n)$ are invertible and the $S$-adic development is everywhere growing, then invariant measures are determined solely by letter cylinder measures.
• The map $\sigma M: \mathcal{M}(X) \to \mathcal{M}(Y)$ induced by a non-erasing morphism $\sigma$ is surjective, preserving measure structure across subshifts.
• The result generalizes to all finite $S$-rank subshifts, where invariant measures are determined by the measures of cylinders $[\sigma_0 \circ \cdots \circ \sigma_{n_0-1}(a_k)]$ for some $n_0$.
• The key insight is that measure rigidity arises from the constancy of the dimension of the letter measure cone across levels in a thin $S$-adic development.