Stable covers of subshifts

TL;DR

This work investigates the existence of characteristic measures for symbolic dynamical systems by examining how the natural SFT cover of a subshift stabilizes. It introduces two stabilization notions, entropy stability and period stability, and proves that both yield characteristic measures; in particular, entropy stability provides a mechanism based solely on entropy differences, while period stability yields a broader class. The authors show that entropy stability is equivalent to language stability, linking two perspectives on stabilization. They also construct a period-stable but language-unstable subshift with no periodic points, and lift the construction to a product with a full shift to obtain a system whose automorphism group contains a non-amenable free group, illustrating that period stability can yield characteristic measures in significantly new settings.

Abstract

Given a dynamical system, a characteristic measure is a Borel probability measure invariant under all of its automorphisms. Frisch and Tamuz asked if every symbolic system supports such a measure. Motivated by this problem, we study the natural cover of a subshift by its shift of finite type approximations and two senses in which this cover can be said to stabilize. The first is in terms of entropy decay and the second in terms of periodic points. We show that the first type of stabilization gives a new characterization of the class of language stable shifts and demonstrates that there is a mechanism for producing a characteristic measures that relies only on entropy differences. For the second type of stabilization, we show that this defines a new class of subshifts, invariant under conjugacies, that have characteristic measures.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 3.1
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['th:main']}
• Theorem 3.3
• Definition 3.4