Gibbs and equilibrium measures for some families of subshifts

TL;DR

This paper generalizes the Lanford-Ruelle theorem to all subshifts, showing that equilibrium measures for functions with $d$-summable variation are topologically Gibbs, and applies this to various interesting subshift families.

Contribution

It extends the Lanford-Ruelle theorem to all subshifts and provides specific proofs for $eta$-shifts, Dyck-shifts, and Kalikow-type shifts.

Findings

Equilibrium measures are topologically Gibbs for all subshifts with $d$-summable variation.

The theorem applies to $eta$-shifts, Dyck-shifts, and Kalikow-type shifts.

Specific proofs are provided for each family of subshifts.

Abstract

For SFTs, any equilibrium measure is Gibbs, as long a $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobru{\v{s}}in states that for strongly-irreducible subshifts, shift-invariant Gibbs-measures are equilibrium measures. Here we prove a generalization of the Lanford-Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is "topologically Gibbs". This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta$-shifts, Dyck-shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford-Ruelle type theorem holds. For each of these families we provide a specific proof of the result.