Conformal measures and the Dobrushin-Lanford-Ruelle equations

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.