Conformal measures and the Dobrušin-Lanford-Ruelle equations

Abstract

We demonstrate the equivalence of two definitions of a Gibbs measure on a subshift over a countable group. We formulate a more general version of the classical Dobrušin-Lanford-Ruelle 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 have the conformal property. We also review methods of constructing an interaction from a potential and vice versa, such that the interaction and the potential have the same Gibbs and equilibrium measures.