Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups

TL;DR

This paper establishes a broad equivalence between relative Gibbs measures and relative equilibrium measures for actions of countable amenable groups, extending classical results to more general settings with applications in symbolic dynamics.

Contribution

It generalizes the Dobrushin-Lanford-Ruelle theorem to relative settings with relaxed constraints, applicable to a wide class of symbolic systems and group actions.

Findings

Equivalence between relative Gibbs and equilibrium measures under general conditions

Extension of Gibbsian characterization to group shifts on countable amenable groups

Applications to pressure maximization and lattice slice equilibrium conditions

Abstract

We formulate and prove a very general relative version of the Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of configuration spaces over a finite alphabet such that for every absolutely summable relative interaction, every translation-invariant relative Gibbs measure is a relative equilibrium measure and vice versa. Neither implication is true without some assumption on the space of configurations. We note that the usual finite type condition can be relaxed to a much more general class of constraints. By "relative" we mean that both the interaction and the set of allowed configurations are determined by a random environment. The result includes many special cases that are well known. We give several applications including (1) Gibbsian properties of measures that maximize pressure among all those that project to a given measure via a topological factor map from one symbolic system to another; (2) Gibbsian properties of equilibrium measures for group shifts defined on arbitrary countable amenable groups; (3) A Gibbsian characterization of equilibrium measures in terms of equilibrium condition on lattice slices rather than on finite sets; (4) A relative extension of a theorem of Meyerovitch, who proved a version of the Lanford--Ruelle theorem which shows that every equilibrium measure on an arbitrary subshift satisfies a Gibbsian property on interchangeable patterns.