Equilibrium measures for two-sided shift spaces via dimension theory

TL;DR

This paper develops a method to describe equilibrium measures for two-sided shift spaces using a Hausdorff measure-like construction, extending previous work to broader classes of systems including subshifts of finite type and shift spaces with synchronizing words.

Contribution

It introduces conditions under which equilibrium measures for shift spaces can be characterized via a Hausdorff measure analogy, generalizing prior applications to hyperbolic systems.

Findings

Applicable to all subshifts of finite type with Hölder continuous potentials

Extended the construction to shift spaces with synchronizing words

Provided a unified framework for equilibrium measures in symbolic dynamics

Abstract

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and H\"older continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.