Conformal measures and the Dobrushin-Lanford-Ruelle equations

TL;DR

This paper establishes the equivalence between conformal measures and Gibbs measures defined by DLR equations on subshifts over countable groups, extending classical concepts through a generalized cocycle framework.

Contribution

It introduces a generalized DLR equation with a measurable cocycle and proves the equivalence with conformal measures, unifying different definitions of Gibbs measures.

Findings

Proves the equivalence of conformal and DLR-based Gibbs measures.

Introduces a generalized DLR framework with measurable cocycles.

Shows how to construct interactions and potentials inducing the same cocycle.

Abstract

We demonstrate the equivalence of two definitions of a Gibbs measure on a subshift over a countable group, namely a conformal measure and a Gibbs measure in the sense of the Dobrushin-Lanford-Ruelle (DLR) equations. We formulate a more general version of the classical DLR equations with respect to a measurable cocycle, which reduce to the classical equations when the cocycle is induced by an interaction or a potential, and show that a measure satisfying these equations must be conformal. To ensure the consistency of these results with earlier work, we review methods of constructing an interaction from a potential and vice versa, such that the interaction and the potential constructed from it, or vice versa, induce the same cocycle.