Stable covers of subshifts

TL;DR

This paper investigates the stabilization of subshift covers by shift of finite type approximations, revealing new classes of shifts with characteristic measures through entropy and periodic point analysis.

Contribution

It introduces two notions of stabilization for subshift covers, characterizes language stable shifts, and defines a new class of subshifts with characteristic measures.

Findings

Entropy-based stabilization characterizes language stable shifts.

Periodic point stabilization defines a new class of subshifts with characteristic measures.

Mechanisms for producing characteristic measures depend on entropy differences.

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.