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

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.

Recoding Lie algebraic subshifts

“…In [19] we proved this equivalence for groups, monoids, semigroups, rings, vector spaces over a finite field, heaps, Boolean algebras and distributive lattices (which are all shallow), and also for quasigroups, loops 1 and lattices (which are not shallow). We are in fact not aware of any varieties where the equivalence fails.…”

“…Weak TMP was defined in [1], where it was also proved that all group shifts on countable groups satisfy it. Mean TMP was defined in the preprint [2] in a more general setting; the above definition is its special case.…”

The Lanford-Ruelle theorem for actions of sofic groups

“…The topological Markov property, as we present it here, was defined in [7]. It was proposed as a generalization of the condition of being a subshift of finite type which was sufficient to prove a generalization of the Lanford-Ruelle theorem for actions of amenable groups.…”

“…From the arguments above and Theorem 4.5 we obtain that equilibrium measures on X are invariant under the action of ∆(X). Using Theorem 6.1, we have the following extension of [7,Corollary 5.4] to groups shifts over sofic groups: Theorem 6.3. Let Γ be a sofic group and let X ⊂ H Γ be a group shift.…”

“…It has been proven that any measure of maximal entropy for a Z d -shift of finite type is invariant with respect to the asymptotic relation [11]. Also, the Lanford-Ruelle theorem has been extended to the context where the acting group is any countable amenable group, not only Z d [31,40], and even more generally, to subshifts which satisfy a condition called the topological Markov property and with respect to any random environment [7]. For an overview of these generalizations, we refer the reader to the introduction of [7].…”