A Dobrushin-Lanford-Ruelle theorem for irreducible sofic shifts

TL;DR

This paper proves that for certain potentials on irreducible sofic shifts, equilibrium measures are exactly the shift-invariant Gibbs measures, extending Gibbsianness preservation results to broader classes of shift spaces.

Contribution

It establishes a Dobrushin-Lanford-Ruelle theorem for irreducible sofic shifts, linking equilibrium measures and Gibbs measures under summable variation potentials.

Findings

Equilibrium measures coincide with Gibbs measures for the specified class.

Gibbsianness preservation extends to finite-to-one factor codes.

The main theorem applies to one-dimensional irreducible sofic shifts.

Abstract

We show that for a potential with summable variations on an irreducible sofic shift in one dimension, the equilibrium measures are precisely the shift-invariant Gibbs measures. The main tool in the proof is a preservation of Gibbsianness result for almost invertible factor codes on irreducible shifts of finite type, which we then extend to finite-to-one codes by applying the results about equilibrium measures.