Thermodynamic formalism for amenable groups and countable state spaces

Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that non-trivially satisfy the conditions for having this equivalence and a non-empty range of inverse temperatures where uniqueness holds.


Introduction
There are two general ways to describe a system composed of many particles: microscopically and macroscopically. The first one makes use of the exact positions of the particles, as well as their local interactions. The second one, in turn, is usually outlined by thermodynamic quantities such as energy and entropy. One could say that statistical mechanics -originated from the works of Boltzmann [10] and Gibbs [32] -is the bridge between the microscopic and the macroscopic descriptions of this kind of systems. In this connection, Gibbs measures are a central object.
It is fair to say that Gibbs measures are at the core of the "conceptual basis of equilibrium statistical mechanics" [52]. Relevant examples are the Ising model, which tries to capture the magnetic properties of certain materials; the hard-core model, that describes the distribution of gas particles in a given environment; among many others [29,30,31]. In these cases it is customary to consider that the many particles interacting are infinite, take a value from a state space A (also called alphabet when A is countable), and they are disposed in a crystalline structure. This structure and its symmetries are usually represented by a countable group G, possibly with some Cayley graph associated with it. A particular case is the hypercubic d-dimensional lattice, which can be understood as the Cayley graph of the finitely generated abelian group G = Z d according to its canonical generators. Then, it is natural to represent an arrangement of particles as an element of the space of configurations X = A G , the G-full shift. Considering this, one is interested in certain measures µ in the space M(X) of Borel probability measures supported on X. More specifically, the measures of interest are the ones that describe these kind of systems when they are in thermal equilibrium, where the energy of configurations is given by some potential φ : X → R. However, there are many mathematically consistent ways to represent that situation by choosing an appropriate measure µ ∈ M(X) and, as the theory evolved, it drew the attention from different areas of expertise such as probability [51,27] and ergodic theory [56,14]. Consequently, the very concept of Gibbs measure started to develop in more abstract and not always equivalent directions.
We focus mainly on four conceptualizations of the idea of thermal equilibrium, namely, DLR, conformal, Bowen-Gibbs, and equilibrium measures. We now proceed to briefly describe each of them.
Dating back to the 60's, Dobrushin [22,23] and, independently, Lanford and Ruelle [41] proposed a concept of Gibbs measure that extended the usual Boltzmann-Gibbs formalism to the infinite particles setting. Roughly, the idea involved looking for probability distributions compatible with a family of maps -sometimes called specification -that prescribe conditional distributions inside finite subsets of G given some fixed configuration outside. More specifically, given a collection γ = (γ K ) K∈F (G) of probability kernels γ K : B × X → [0, 1], with F(G) the set of finite subsets of G and B the Borel σ-algebra, one is interested in finding measures µ ∈ M(X) such that µγ K = µ for every K ∈ F(G), where µγ K is a new measure (a priori, different from µ) obtained from µ via γ K . Those distributions are called DLR measures after the above cited authors and they have received considerable attention from both mathematical physicists and probabilists (see, for example, [30,31,38,52]).
Another rather classical way to define a Gibbs measure, which does not involve conditional distributions, was introduced by Capocaccia in [17]. Given a class E of local homeomorphisms τ : X → X and a potential φ : X → R, one is interested in measures µ such that d(µ•τ −1 ) dµ = exp(φ τ * ) for every τ ∈ E, where φ τ * : X → R is a function representing the energy difference between a configuration x and τ (x) (e.g., see [38,Definition 5.2.1]). This kind of measures fits in the more general context of (Ψ, R)-conformal measures explored in [1], where R is a Borel equivalence relation and Ψ : R → R + is a measurable function. Then, Capocaccia's measures, that we simply call conformal measures, can be recovered by taking a function Ψ related to the given potential and R the tail relation in the space of configurations. By considering other particular Borel relations R and measurable functions Ψ, one can recover other relevant notions of conformal measures, such as the ones presented in [20,49,53], that are mainly adapted to the one-dimensional setting, i.e., when G = Z or, considering also semigroups, when G = N.
A third possibility, introduced by Rufus Bowen in a one-dimensional and ergodic theoretical context [14], is to define Gibbs measures by specifying bounds for the probability of cylindrical events. More concretely, one is interested in the measures µ ∈ M(X) for which there exists constants C > 0 and p ∈ R such that C −1 ≤ µ([a 0 a 1 · · · a n−1 ]) exp( As in [7], we call those measures Bowen-Gibbs measures to avoid confusion. This definition has been considered in the literature [18,36,38,52] and also relaxed versions of it, such as the so-called weak Gibbs measures [58,60], where the constant C is replaced by a function that grows sublinearly in n. This and further relaxations have also played a relevant role in the multi-dimensional case, this is to say, when G = Z d and d > 1, for finite state spaces (e.g., see [38,Theorem 5.2.4]).
The last important definition considered in this work is the one of equilibrium measure. When X is a finite configuration space, equilibrium measures are simply probability vectors that maximize the sum (or difference) of an entropy-and an energy-like quantity, that is, a quantity like where k = |X|, x i ∈ X, φ : X → R is a potential, p = (p 1 , . . . , p k ) is a probability vector with p i the probability associated with x i , and H(p) is the Shannon entropy of p. These measures were considered, for example, in [31,38,52]. On the other hand, when X is an infinite configuration space and there is a robust notion of specific entropy, let's say h(µ), we are interested in studying measures µ ∈ M(X) that maximize the quantity h(µ) +´φ dµ for a continuous potential φ : X → R. This notion tries to capture the macroscopic behaviour of the system without making any assumption of the microscopic structure.
The problem of proving equivalences among these and other related notions has already been studied in different settings. We mention some relevant results that can be found in the literature.
In the one-dimensional case, for finite state spaces, Meyerovitch [44] proved the equivalence between conformal measures and DLR measures for some families of proper subshifts. Also, Sarig [54,Theorem 3.6] proved that any DLR measure on a mixing subshift of finite type is a conformal measure, for a different but related notion of conformal, restricted to the one-dimensional setting. In the same work, for one-sided and countably infinite state spaces, Sarig [54,Proposition 2.2] proved that conformal measures -according to his definition -are DLR measures for topological Markov shifts. In this same setting, Mauldin and Urbański [43] proved the existence of equilibrium measures and that any equilibrium measure satisfies a Bowen-Gibbs equation. Moreover, if the topological Markov shift satisfies the BIP property and the potential has summable variation, Beltrán, Bissacot, and Endo [6] proved that DLR measures and conformal measures -in the same sense as Sarig -are equivalent. Finally, for potentials with summable variation on sofic subshifts, Borsato and MacDonald [12] proved the equivalence between DLR and equilibrium measures. There are also other classes of measures in the one-dimensional case which we do not treat here, such as g-measures [37,59] and eigenmeasures associated with the Ruelle operator [14,52]. When the state space is finite, it is known that the set of DLR measures and g-measures do not contain each other [28,9], but there is a characterization for when a g-measure is a DLR measure [7]. In addition, eigenmeasures coincide with DLR measures for continuous potentials in the one-sided setting, as proven by Cioletti, Lopes, and Stadlbauer in [19]. Pioneering works in the one-dimensional countably infinite state space setting can be found in [33,34].
In the multi-dimensional case, some results regarding the equivalences among the four notions of Gibbs measures have been proved for finite state spaces. A first important reference is Keller [38,Theorem 5.2.4 and Theorem 5.3.1], where it is proven that when φ : X → R is regular (which includes the case of local and Hölder potentials, and well-behaved interactions), then the four definitions are equivalent. Here, by regular, we mean that where δ n (φ) is the oscillation of φ when considering configurations that coincide in a specific finite box, namely, [−n, n] d ∩ Z d . Other classical references in this setting are due to Dobrushin [21] and Lanford and Ruelle [41], which, combined, establish the equivalence between DLR measures and equilibrium measures for a general class of subshifts of finite type. Kimura [40] generalized the equivalence between DLR and conformal measures for subshifts of finite type, and some of the implications are true for more general proper subshifts. In the countably infinite state space setting, Muir [45,46] obtained all equivalences for the G-full shift when G = Z d . In order to do this, it was required that the potential φ : X → R is regular and satisfies a normalization criterion, namely, exp-summability: This last condition is automatically satisfied when A is finite.
Results proving equivalences between different kinds of Gibbs measures go beyond the amenable [55,5,2,15] and even the symbolic setting to general topological dynamical systems [3,36].
One of our main contributions is to exhibit conditions to guarantee that the four notions of Gibbs measures presented above are equivalent, when considering the state space A = N and an arbitrary countable amenable group G, thus extending Muir's methods to the more general amenable case. Countable amenable groups play a fundamental role in ergodic theory [48] and include many relevant classes of groups, such as abelian (so, in particular, G = Z d ), nilpotent, and solvable groups and are closed under many natural operations, namely, products, extensions, etc. (e.g., see [42]). In the more general group and finite state space setting, the equivalence between DLR and conformal measures was extended to general subshifts over a countable discrete group G with a special growth property by Theorems 5 and Theorem 6]. Recently, a different proof for the equivalence between DLR and conformal measures for any proper subshift was given by Pfister in [50]. Also, in [4], a Dobrushin-Lanford-Ruelle type theorem is proven in the case that the group is amenable and a topological Markov property holds, which is satisfied, in particular, by subshifts of finite type. Here, as Muir, we focus on the G-full shift case. We consider the configuration space X = N G , for G an arbitrary countable amenable group, and an exp-summable potential φ : X → R with summable variation (according to some exhausting sequence). The concept of summable variation extends the one of regular potential presented before. More precisely, a potential φ has summable variation if where {E m } m is an exhausting sequence for G and δ Em (φ) is a standard generalization of δ m (φ). The paper is organized as follows. First, in Section 2, we present some preliminary notions about amenable groups G, the corresponding symbolic space N G , and potentials. Later, in Section 3, we introduce the concept of pressure in our framework and we prove its existence. Also, we prove that it satisfies an infimum rule and that it can be obtained as the supremum of the pressures associated with finite alphabet subsystems. In order to achieve this, we use relatively new techniques for tilings of amenable groups [26] and, inspired by ideas for entropy from [25], we develop a generalization of Shearer's inequality for pressure. In Section 4, we introduce spaces of permutations and Gibbsian specifications in order to pave the way for the definitions of conformal and DLR measures, respectively. Next, in Section 5, we prove the equivalence between the four notions of Gibbs measures mentioned above given some conditions on the potential, such as exp-summability and summable variation. We also prove related results involving equilibrium measures. In order to prove the equivalence between DLR and conformal measures we rely on the strategies presented on [45] for the G = Z d case, which already considers an infinite state space. Moreover, using Prokhorov's Theorem and relying on the existence of conformal measures in the compact setting [20], we prove the existence of a conformal (and DLR) measure in our context. We also prove that DLR measures are Bowen-Gibbs. If it is also the case that the measure is invariant under shift actions of the group, we prove that any Bowen-Gibbs measure is an equilibrium measure and that any equilibrium measure is a DLR measure. At last, in Section 6, we show how to recover previous results from ours and, inspired by the Potts model and considering a version of it with countably many states, we exhibit a family of examples for which all our results apply non-trivially and, in addition, a version of Dobrushin's Uniqueness Theorem adapted to our setting holds, thus providing a regime where the uniqueness of a Gibbs measure is satisfied.

Preliminaries
2.1. Amenable groups and the space N G . Let G be a countable discrete group with identity element 1 G and N be the set of non-negative integers. Consider the G-full shift over N, that is, the set N G = {x : G → N} of N-colorings of G, endowed with the product topology. We abbreviate the set N G simply by X. Given a set A, denote by F(A) the set of nonempty finite subsets of A.
Consider a sequence {E m } m of finite sets of G such that E 0 = ∅, 1 G ∈ E 1 , E m ⊆ E m+1 for all m ∈ N, and m∈N E m = G. We will call such a sequence an exhaustion of G or an exhausting sequence for G. Throughout this paper, we will consider a particular type of exhausting sequences: we will assume further that Given a fixed exhaustion {E m } m , the topology of X is metrizable by the metric d : X × X → R given by d(x, y) = 2 − inf{m∈N : x Em =y Em } , where x F denotes the restriction of a configuration x to a set F ⊆ G. Denote by X F = {x F : x ∈ X} the set of restrictions of x ∈ X to F . The sets of the form [w] = {x ∈ X : x F = w}, for w ∈ X F , F ∈ F(G), are called cylinder sets. The family of such sets is the standard basis for the product topology of N G .
Let B be the σ-algebra generated by the cylinder sets and let M(X) be the space of probability measures on X. Consider also M G (X) the subspace of G-invariant probability on X.
The group G acts by translations on X as follows: for every x ∈ X and every g, h ∈ G, This action is also referred, in the literature, as the shift action. Moreover, it can be verified that , for every x ∈ X, g ∈ G, and F ⊆ G.

Potentials and variations.
A function φ : X → R is called a potential. Given E ⊆ G, the variation of φ on E is given by Let {E m } m be an exhausting sequence for G. Given a potential φ : X → R, it is not difficult to see that φ is uniformly continuous if and only if lim m→∞ δ Em (φ) = 0. In this context, given F ∈ F(G), we define the F -sum of variations of φ (according to {E m } m ) as Remark 1. For any exhausting sequence {E m } m and any F ∈ F(G), the sequence Proposition 2.1. Let φ : X → R be a potential such that the F -sum of variation of φ is finite for some F ∈ F(G). Then φ is a uniformly continuous potential.
Proof. Let {E m } m be an exhausting sequence for G. Since, in particular, E m ⊆ E m+1 for every m ≥ 1, we have that 0 ≤ δ Em+1 (φ) ≤ δ Em (φ) for every m ≥ 1. Then, for every M ≥ 1, where the last line follows from Remark 1. Therefore, and the result follows.

Definition 1.
Let ϕ : F(G) → R be a function. Given L ∈ R, we say that ϕ(F ) converges to L as F becomes more and more invariant if for every > 0 there exist K ∈ F(G) and δ > 0 such that |ϕ(F ) − L| < for every (K, δ)-invariant set F ∈ F(G). We will abbreviate such a fact as lim For example, if G = Z d and F n = [−n, n] d ∩ Z d , then {F n } n is a Følner sequence for Z d . It is not difficult to see that if lim F →G ϕ(F ) = L, then lim n→∞ ϕ(F n ) = L for every Følner sequence {F n } n . In particular, when G = Z d , convergence as F becomes more and more invariant implies convergence along d-dimensional boxes, which is a common condition in the multi-dimensional framework. It is not difficult to see that a group is amenable if and only if it has Følner sequence. Moreover, for every amenable group G there exists a Følner sequence that is also an exhaustion.

Proposition 2.2.
Let φ : X → R be a potential with summable variation according to an exhausting sequence {E m } m . Then, Proof. Let > 0. Since φ : X → R has summable variation, there exists m 0 ≥ 1 such that Then, for every F ∈ F(G), Due to the amenability of G, for any given m 0 ≥ 1, we have that, for all m ≤ m 0 , Therefore, for every > 0, there exists m 0 ≥ 1 and K ∈ F(G) such that for every (K, )-invariant set F , where C = 1 + V (φ). Since was arbitrary, we conclude.
Given a potential φ : Lemma 2.3. Let {E m } m be an exhausting sequence for G, φ : X → R be a potential that has finite oscillation and such that lim inf m→∞ δ Em (φ) = 0. Then, In particular, if φ has summable variation according to an exhausting sequence {E m } m , then equation (1) holds.
Proof. Let > 0. Since lim inf m→∞ δ Em (φ) = 0, there exists m 0 ≥ 1 such that δ Em 0 (φ) ≤ . Denote E m0 by K. Due to amenability, we can find Considering this, if x, y ∈ X are such that x F = y F , we have that and the result follows.

Pressure
We dedicate this section to introduce the pressure of a potential. We define and work on the setting of exp-summable potentials with summable variation on a countable alphabet. The pressure -basically equivalent to the specific Gibbs free energy -is a very relevant thermodynamic quantity that helps to capture the concept of Gibbs measure in a quantitative way.
First, we prove that the pressure, which we define through a limit over sets that are becoming more and more invariant, exists in the finite alphabet case. The definition of the pressure is often done in terms of a particular Følner sequence, which is an, a priori, less robust and less overarching approach. Existence of the limit for a particular Følner sequence {F n } n and the fact that it is independent on the choice of such sequence is well-known (see, for example, [57,35,16], in the context of absolutely summable interactions). Here, we prove something stronger: that our definition of pressure obeys the infimum rule -which is a refinement of the Ornstein-Weiss Lemma (see, for example, [39, §4.5]) -, this is to say, it can be expressed as an infimum over all finite sets of G. In order to conclude this, we extend the results about Shearer's inequality in [25] for topological entropy to pressure. Now, in the countable alphabet context, we take a similar approach. First, we consider again a definition of pressure in terms of sets that are becoming more and more invariant. Next, we prove that the infimum rule still holds and, finally, we prove that the pressure can be obtained as the supremum of the pressures associated with finite alphabet subsystems. A related result was obtained by Muir in [45] for the Z d group case, where the pressure was defined as a limit over a particular type of Følner sequence, namely, open boxes centered at the origin of radius n. The existence of this limit was proven through a sub-additivity argument that exploits the property that large boxes can be partitioned into many equally sized ones, which might not be valid in more general groups. In order to generalize this idea of partitioning sets, we make use of tiling techniques introduced in [26], which, together to what is done in the finite alphabet case, allow us to prove the infimum rule for infinite alphabets over a countable amenable group. This type of result was not considered in [45].
We begin by introducing some definitions. Given a potential φ : X → R and F ∈ F(G), define the partition function for φ on F as We define the pressure of φ, which we denote by p(φ), as whenever such limit as F becomes more and more invariant exists. In addition, given a finite subset A ∈ F(N), we define Z F (A, φ) as the partition function associated with the restriction of φ to A G . More precisely, Similarly, we define p(A, φ) as whenever such limit exists.
3.1. Infimum rule for finite alphabet pressure. The main goal of this subsection is to prove the following theorem.
In order to prove this result, we require some definitions. A function ϕ : A k-cover K of a set F ∈ F(G) is a family {K 1 , K 2 , . . . , K r } ⊆ F(G) (with possible repetitions) such that each element of F belongs to K i for at least k indices i ∈ {1, . . . , r}. We say that ϕ satisfies Shearer's inequality if for any F ∈ F(G) and any k-cover K of F , it holds that Notice that Shearer's inequality implies sub-additivity. Considering this, we have the following key lemma.
Moreover, if ϕ satisfies Shearer's inequality, then In this last case, we say that ϕ satisfies the infimum rule. Now, fix a finite alphabet A ∈ F(N). For a continuous potential φ : X → R, we denote by φ A the supremum norm of φ over the compact set X ∩ A G , i.e., φ A = sup x∈X∩A G |φ(x)|. Next, given a set E ⊆ G, F ∈ F(G), and u E ∈ X E ∩ A E , we define where the supremum is over Then, it is easy to check that for any E ⊆ G and u E ∈ A E , the functionφ : Next, consider the function ϕ : F(G) → R defined as ϕ(F ) = log Z F . From the properties above and properties of the log(·) function, it follows that ϕ is non-negative and monotone. Moreover, ϕ is Ginvariant. The following lemma is a generalization of [25, Lemma 6.1] designed to address the pressure case instead of just the topological entropy and, in particular, it claims that ϕ satisfies Shearer's inequality.
Proof. Given a k-cover K of F , notice that, since φ| X∩A G is non-negative, for any x ∈ X ∩ A G . We proceed by induction on the size of F \ E. First, suppose that |F \ E| = 0. Then, F \ E = ∅ and with |F \ E| ≤ n, and every k-cover K of F . We will show that the same holds for E, F with |F \ E| = n + 1. Fix g ∈ F \ E and notice that |F \ (E ∪ {g})| = n. Then, Notice that the first inequality follows from the induction hypothesis and the third inequality follows from the generalized Hölder inequality. Indeed, consider p ≤ 1 such that K∈K:g∈K . By the generalized Hölder inequality, where K∈K:g∈K and, since · p is monotonically decreasing in p for any fixed |A|-dimensional vector, Proof (of Theorem 3.1). As a consequence of Lemma 3.3, we have that if φ| X∩A G is non-negative, then ϕ satisfies Shearer's inequality. Thus, by the Ornstein-Weiss lemma, p(A, φ) exists and it satisfies the infimum rule, i.e., Finally, in order to deal with the general case, it suffices to apply the previous result to φ + φ and then observe that Remark 2. Notice that the previous results (namely, Lemma 3.3 and Theorem 3.1) also hold for G-subshifts, this is to say, any closed and G-invariant subsets X of N G .

3.2.
Tilings. Pressure is one of the most important notions in thermodynamic formalism. One key technique to properly define pressure is sub-additivity, which is based on our ability to partition a system in smaller and representative pieces. In the context of countable amenable groups, it appears to be necessary to generalize tools that are classically used in the Z d case (e.g., [52,45]). In order to do this, we will begin by exploring the concept of (exact) tilings of amenable groups and the relatively recent techniques introduced in [26].  We say that a sequence {T n } n of tilings is congruent if, for each n ≥ 1, every tile of T n+1 is equal to a (disjoint) union of tiles of T n . The following theorem is the main result in [26], which gives sufficient conditions so that we can guarantee the existence of such sequence with extra invariance properties. Given a tiling T , we define S T = S∈S(T ) SS −1 . Notice that S T contains every shape S ∈ S(T ), S −1 T = S T , and 1 G ∈ S T . Given a tiling, the next lemma provides a way to approximate any sufficiently invariant shape by a union of tiles. Lemma 3.5. Given K ∈ F(G) and δ > 0, consider a tiling T with (K, δ)-invariant shapes. Then, for any > 0 and any Proof. Consider a tiling T made of (K, δ)-invariant shapes and > 0. Therefore, where we have used that

Infimum rule for countable alphabet pressure. We say that
. Finally, observe that if φ is exp-summable, then it must be bounded from above.
Before stating the main result of this section, we begin by the next lemma, that guarantees that given a finite shape F , one can approximate the partition function on F using a finite alphabet. Lemma 3.6. Let φ : X → R be an exp-summable and uniformly continuous potential. Then, for every > 0 and every On the other hand, since φ is uniformly continuous, there must be an index m ≥ 1 for which In addition, for each such w , pick a configuration x w ∈ [w ] such that Hence, The next proposition establishes a fundamental connection between the partition function for sufficiently invariant sets F ∈ F(G) and the pressure for a sufficiently large finite alphabet A. Proposition 3.7. Let φ : X → R be an exp-summable and uniformly continuous potential with finite oscillation. Then, for every 1 2 By Theorem 3.4, there exists a tiling T such that its shapes are (K , δ )-invariant. Without loss of generality, by possibly readjusting K and δ , assume that |S | ≥ − 1 log(1 − ) for every S ∈ S(T ). Therefore, by Lemma 3.6, for every exists, so we can pick K ∈ F(G) and δ > 0 such that K ⊇ K , δ < δ , and Next, by Theorem 3.4, we can obtain a tiling T of (K, δ)-invariant sets such that every tile in T is a union of tiles in T , i.e., S = S ∈S(T ) c ∈C S (S ) S c . Furthermore, by Lemma 3.5, for every Furthermore, for every S ∈ S(T ), we have that where we used that, for every g ∈ G, Therefore, from equation (3), we obtain that where the second inequality follows from equation (5) and the third from equation (4). Hence, if In addition, We conclude that, for every 0 < < 1 2 , there exist A ∈ F(N), K ∈ F(G), and δ > 0 such that for Since was arbitrary, we conclude the result.
Now we can prove the following generalization of Theorem 3.1.
Theorem 3.8. Let φ : X → R be an exp-summable and uniformly continuous potential with finite oscillation. Then, p(φ) exists and Proof. By Proposition 3.7, for every 1

Permutations and specifications
In order to define conformal and DLR measures it will be crucial to introduce coordinate-wise permutations and specifications. We begin by describing and exploring some properties of coordinatewise permutations.
4.1. Coordinate-wise permutations. Let S N be the set of all permutations of N. Following [38,45], we now introduce a class of local maps on X. Given an exhausting sequence {E m } m , this class will allow us to understand how φ Em (x) behaves if x is changed at finitely many sites and it will be central when defining conformal measures in §5.
where τ g ∈ S N . We usually denote τ by τ K to emphasize the set K.
Let E = K∈F (G) E K and notice that there is a natural action of G on E given by where g ∈ G, x ∈ X, K ∈ F(G), τ K ∈ E K , and g · τ K ∈ E Kg −1 . In order to avoid ambiguity, we will denote g · τ K by τ Kg −1 and that will be enough for our purposes.
We can also restrict ourselves to permutations over a finite alphabet. More explicitly, for A ∈ F(N) and K ∈ F(G), define Notice that E is a group with the composition generated by single-site permutations τ g , where E K and E K,A are subgroups. Moreover, observe that if g = h, then τ g τ h = τ h τ g . We will also consider a particular type of permutations, which are defined below.

Definition 4.
Given K ∈ F(G) and w, w ∈ X K , let τ w,w : X → X be the map defined as It is clear that τ w,w ∈ E K , τ w,w = τ w ,w and that τ w,w is an involution, that is, it is its own inverse. .
Proof. Let K, E, F ∈ F(G) and τ K ∈ E K be as in the statement of the Lemma. Then, it is easy to verify that, for any x ∈ X, Given a potential φ : X → R with summable variation according to an exhausting sequence {E m } m , the next theorem tells us that the asymptotic behaviour of φ Em (x) is essentially independent of the value of the configuration x at finite sets K ∈ F(G). The reader can compare the next result with [ exists uniformly on X and on E K . Moreover, such limit does not depend on the exhausting sequence.
Proof. First, suppose that K is a singleton {h} for some h ∈ G and let > 0. Since φ has summable variation according to Therefore, for every > 0, there exists m 1 ≥ m 0 such that for every m ≥ m 1 , there exists k m such that for every k ≥ k m , Notice that, in the particular case that {Ẽ m } m is the same as {E m } m , one just need to take k m = m and the same inequality would follow. This proves that {φ τ h Em } m is a Cauchy sequence for any τ h ∈ E {h} , which implies that the uniform limit i.e., the limit is independent of the exhausting sequence provided φ has summable variation according to some exhausting sequence. Now, let's consider a general K ∈ F(G) and write where we regard τ ∅ as the identity, so the first equality follows from the fact that the considered sum is telescopic. Therefore, by considering the uniform convergence for singletons, which concludes the result.

Corollary 4.3.
Let φ : X → R be a potential with summable variation according to an exhausting sequence {E m } m . Then, for all K ∈ F(G) and for all τ K ∈ E K , Proof. Notice that, given g ∈ G and x ∈ X, we have that since {E m g} m is also an exhausting sequence.
for every m ∈ N such that F ⊆ E m . Therefore, by Theorem 4.2,

Specifications.
This section tackles results about specifications, a concept related to DLR measures. More precisely, DLR measures can be defined using a special kind of specifications, but here we begin by presenting some more general results. Let B be the Borel σ-algebra, that is, the σ-algebra generated by the cylinder sets, and, for each K ∈ F(G), let B K be the σ-algebra generated by cylinder sets [w], with w ∈ X K . Now, a specification in our context, will mean a family γ = (γ K ) K∈F (G) of maps γ K : B × X → [0, 1] such that i) for each x ∈ X, the map B → γ K (B, x) is a probability measure on M(X); ii) for each B ∈ B, the map In other words, γ is a particular family of proper probability kernels that satisfies consistency condition (iv). An element γ K in the specification maps each µ ∈ M(X) to µγ K ∈ M(X), where and each B-measurable function h : X → R to a B K c -measurable function γ K h : X → R given by It can be checked that (µγ K )(h) = µ(γ K h). The probability measures on the set K (B, ·) µ-a.s., for all B ∈ B and K ∈ F(G)} are said to be admitted by the specification γ. Now, we restrict ourselves to a particular kind of specification. Namely, given an exhausting sequence of finite sets {E m } m and φ : X → R an exp-summable potential with summable variation according to {E m } m , consider γ = (γ K ) K∈F (G) the specification coming from φ, where each γ K : B × X → [0, 1] is given by , for each B ∈ B and x ∈ X. The collection γ is a (Gibbsian) specification. The expression in equation (7) is well-defined due to the following proposition.
Proposition 4.6. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . If K ∈ F(G), the limit ) exists for each w ∈ X K , uniformly on X. Furthermore, for every B ∈ B and every x ∈ X, it holds that In order to prove Proposition 4.6, we require two lemmas, which we state and prove next.
Lemma 4.7. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, for any K ∈ F(G) and for any m ∈ N such that K ⊆ E m , Proof. Let K ∈ F(G) and x, y ∈ X be such that x G\K = y G\K . Notice that for any g ∈ G, (g · x) G\Kg −1 = (g ·y) G\Kg −1 . In addition, given m ∈ N, we have that . Considering this, we obtain Now, let m 0 ∈ N be the smallest index such that K ⊆ E m0 . Then, for every m ≥ m 0 , every x ∈ X, and every v, w ∈ X K , we have that and, similarly, Lemma 4.8. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, for any K ∈ F(G) and w ∈ X K , uniformly on X.
Proof. Given K ∈ F(G), w ∈ X K , and x ∈ X, consider the sequence of functions f m : , uniformly on X. In addition, by Lemma 4.7, there exist m 0 ∈ N and a constant C = exp(∆ K (φ) + V K (φ)) > 0 such that for every m ≥ m 0 and for every v ∈ X K , so h (and therefore, C · h) is integrable with respect to the counting measure in X K . Therefore, by the Dominated Convergence Theorem, if follows that Proof (of Proposition 4.6). First, note that for any given K ∈ F(G), for all m ∈ N and, due to Lemma 4.8, the left-hand side is bounded away from zero uniformly in m.
. Therefore, uniformly on X, again due to Lemma 4.8. Now, let B ∈ B and x ∈ X. Then, uniformly on X, where the exchange of the limit and the sum follows from Lemma 4.8. Proposition 4.9. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, for every K ∈ F(G), w ∈ X K , and x ∈ X, the equation where the last equality follows from Theorem 4.2. Also, if m 0 ∈ N is such that K ⊆ E m0 , the exchange of limit and sum in the denominator from the first to the second line follows from Lemma 4.8.

Corollary 4.10.
Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, for every K ∈ F(G), γ K is G-invariant, that is, for every w ∈ X K , x ∈ X, and g ∈ G, it holds that Proof. Let K ∈ F(G). Given v ∈ X K , let y v ∈ X be arbitrary and such that y v K = v. Then, where we have used the property of φ τ * from Corollary 4.3.

Definition 6. A specification
Remark 4. In order to verify that a specification is quasilocal it suffices to prove that γ K h ∈ L, for K ∈ F(G) and h ∈ L (see [30], page 32).

Theorem 4.11. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence
is defined as in equation (7), then γ is quasilocal.
Proof. Let h ∈ L and let > 0. Given any K ∈ F(G), first notice that In addition, if x, y ∈ X are such that x En = y En for n to be determined, we have that To justify the second inequality, first observe that, for every w, v ∈ X K , φ τv,w * is uniformly continuous, since it is a uniform limit of uniformly continuous potentials, namely, φ Em . Then, there exists n 0 ∈ N such that for every n ≥ n 0 , every w, v ∈ X K , and every x, y ∈ X with x En = y En , Now, since h is local, we have that lim n→∞ sup x,y∈X x En =y En |h(x)−h(y)| = 0, so that there exists n 1 ∈ N such that for all n ≥ n 1 , sup x,y∈X x En =y En |h(x) − h(y)| < . Taking n = max{n 0 , n 1 }, we obtain that and since was arbitrary, we conclude.

Equivalences of different notions of Gibbs measures
In this section, we introduce the four notions of Gibbs measures to be considered, namely, DLR, conformal, Bowen-Gibbs, and equilibrium measures, and prove the equivalence among them provided extra conditions. We mainly assume that G is a countable amenable group, the configuration space is X = N G , and φ : X → R is an exp-summable potential with summable variation according to an exhausting sequence {E m } m .
We proceed to describe the content of each subsection: in §5.1, we provide a rigorous definition of each kind of measure and results about entropy and pressure; in §5.2, we establish that the set of DLR measures and the set of conformal measures coincide; in §5.3, we prove that every DLR measure is a Bowen-Gibbs measure; in §5.4, we show the existence of a conformal measure; in §5.5, we prove that a G-invariant Bowen-Gibbs measure with finite entropy is an equilibrium measure; finally, in §5.6, we prove that if a measure is an equilibrium measure, then it is also a DLR measure.
Below, we provide a diagram of the main results of this section, including extra assumptions needed.
DLR measure k s Remark 5. We are not aware whether it is possible to prove that a Bowen-Gibbs measure is necessarily a DLR measure without the finite entropy assumption. In fact, we do not know if G-invariance is a necessary assumption for that implication.

Definitions of Gibbs measures.
We start by giving the definitions of DLR, conformal, and Bowen-Gibbs measures.

Definition 7.
Let φ : X → R be an exp-summable potential with summable variation according to for every K ∈ F(G), B ∈ B, and x ∈ X, where γ K is defined as in equation (7). We denote the set of DLR measures for φ by G(φ).

Definition 8.
Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . A measure µ ∈ M(X) is a conformal measure (for φ) if for every A ∈ F(N), K ∈ F(G), and τ ∈ E K,A .

Definition 9.
Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . A measure µ ∈ M(X) is a Bowen-Gibbs measure (for φ) if there exists p ∈ R such that, for every > 0, there exist K ∈ F(G) and δ > 0 such that, for every (K, δ)-invariant set F ∈ F(G) and x ∈ X, Remark 6. Notice that, in Definition 9, we can replace φ F (x) by sup φ F ([x F ]) in an equivalent way, so that we have Proof. Indeed, given > 0, there exist K ∈ F(G) and δ > 0 so that for every (K, δ)-invariant set F ∈ F(G) and every x ∈ X. Since x is arbitrary, we have that and, since µ is a probability measure, adding over all x F ∈ X F , we get Then, if we take logarithms and divide by |F |, we obtain that so, taking the limit as F becomes more and more invariant, we obtain that and since was arbitrary, we conclude that p = p(φ).
Consider the canonical partition of X given by {[a]} a∈N . This is a countable partition that generates the Borel σ-algebra B under the shift dynamic. Given a measure ν ∈ M(X), the Shannon entropy of the canonical partition associated with ν is given by

([a]) log ν([a]).
Now, for each F ∈ F(G), let {[w]} w∈X F be the F -refinement of the canonical partition and consider its corresponding Shannon entropy, which is given by We have the following proposition. Proposition 5.2. Let φ : X → R be an exp-summable and continuous potential with finite oscillation.
Proof. Let {A n } n an exhausting sequence of finite alphabets and F ∈ F(G). Consider X F,n = {x ∈ X : x F ∈ A F n } ∈ B F . Since φ is exp-summable, then it is bounded from above. Without loss of generality, suppose that it is bounded from above by 0. Thus, so is φ F . Define x ∈ X F,n ; 0, otherwise.
Notice that, for every x ∈ X, φ F (x) = lim n→∞ φ F,n (x) and, for every n ∈ N, φ F (x) ≤ φ F,n+1 (x) ≤ φ F,n (x). Therefore, by the Monotone Convergence Theorem, we can conclude that ). Then, lim n→∞ H F,n (ν) = H F (ν). Also, for each n ∈ N and F ∈ F(G), notice that φ F,n ≤ w∈A F n 1 [w] sup φ F ([w]). Therefore, for every n ∈ N and F ∈ F(G), where we assume that all the sums involved are over cylinder sets with positive measure. The second inequality follows from Jensen's inequality. In addition, notice that, in the case that ν is G-invariant, it follows that where we have used that log Z F (φ) ≤ |F | log Z 1 G (φ) and´φ F dν = |F |´φdν. Therefore, H F (ν) < ∞ and, in particular, Through a standard argument (for example, for the case G = Z, see [24]; the general case is analogous), it can be justified that if the canonical partition has finite Shannon entropy, the Kolmogorov-Sinai entropy of ν can be written as The next proposition is based on [45, Lemma 4.9] and gives us an upper bound in terms of the pressure for the specific Gibbs free energy of a given measure with respect to some potential. Sometimes this fact is known as the Gibbs inequality.

Definition 10.
Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . A measure µ ∈ M G (X) is an equilibrium measure (for φ) if φdµ > −∞ and Notice that it is not clear whether the supremum in equation (11) is achieved. The answer to this problem is intimately related to the concept of Gibbs measures in its various forms and their equivalences, which we address throughout this section.
Remark 7. Notice that, in light of Proposition 5.2, any measure ν ∈ M G (X) such that the´φdν > −∞ has finite entropy, that is, h(ν) < ∞, provided that φ is exp-summable and has finite oscillation. Thus, in the particular case that φ is an exp-summable potential with summable variation according to an exhausting sequence {E m } m , we obtain that h(ν) < ∞.

Equivalence between DLR and conformal measures.
This section is dedicated to proving that the notions of DLR measure and conformal measure coincide in the full shift with countable alphabet over a countable amenable group context. Nevertheless, before proving this major result, notice that for B ∈ B, K ∈ F(G), and x ∈ X, Indeed, it can be checked that This observation will allow us to reduce our calculations from arbitrary Borel sets B ∈ B to cylinder sets of the form [w]. Next, we have the following result.
Corollary 5.4. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . A measure µ ∈ M(X) is a DLR measure for φ if, and only if, for every K ∈ F(G), w ∈ X K and x ∈ X, it holds that Proof. If µ is a DLR measure for φ, then for every K ∈ F(G), B ∈ B, and x ∈ X, Thus, in particular, if w ∈ X K , it holds that and the result follows from Proposition 4.9.
On the other hand, if we assume that for every K ∈ F(G), w ∈ X K , and x ∈ X, equation (13) holds µ(x)-almost surely for every v ∈ X K , then, from equation (12) and Proposition 4.9, µ(x)-a.s it holds that In order to relate the functions γ K that appear in the definition of DLR measures with the permutations involved in the definition of conformal measures, we have the following lemma. Lemma 5.5. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, for every K ∈ F(G), v, w ∈ X K and τ ∈ E K such that x). Proof. Indeed, by Proposition 4.9, for every x ∈ X, Proof. Suppose first that µ ∈ M(X) is a conformal measure for φ and let K ∈ F(G). Begin by noticing that if B ∈ B K c and, for some w ∈ X K and x ∈ X, wx K c ∈ B, then vx K c ∈ B, for every v ∈ X K . As a consequence, we have that, for all τ ∈ E K and all On the other hand,ˆB Therefore, for any w, v ∈ X K , µ(x)-almost surely it holds that Now, let A ∈ F(N) be a finite alphabet and v ∈ A K . For any w ∈ A K , we have that τ w ,v ∈ E K,A . Summing equation (14) over all w ∈ A K , we obtain that µ(x)-almost surely it holds that If {A n } n is an exhausting sequence of finite alphabets, then n≥1 A K n × X K c c = ∅. Moreover, for , µ(x)-almost surely in L 2 (µ) as n → ∞. Therefore, for any fixed v ∈ X K , there exists n 0 ∈ N be such that v ∈ A K n0 and, consequently, v ∈ A K n , for all n ≥ n 0 . Therefore, µ(x)-almost surely it holds that Moreover, equation (14) yields that, for any w ∈ X K , µ(x)-almost surely so that, for any w ∈ X K , µ(x)-almost surely it holds that Therefore, due to Corollary 5.4, µ is a DLR measure. Conversely, suppose that µ ∈ M(X) is a DLR measure for φ and let A ∈ F(N), K ∈ F(G), and τ ∈ E K,A . For any v ∈ X K and w = τ −1 ([v]) K , due to Lemma 5.5, we obtain which concludes the result.

DLR measures are Bowen-Gibbs measures.
This subsection is dedicated to proving that, provided some conditions, any DLR measure for a potential φ is a Bowen-Gibbs measure for φ.
Proposition 5.7. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . If µ ∈ M(X) is a DLR measure for φ, then, for every F ∈ F(G), w ∈ X F and y ∈ X, it holds µ(x)-almost surely that Proof. Let F ∈ F(G) and τ ∈ E F . From Proposition 4.4, we have that for every x ∈ X, , which, in particular, yields that, for every x ∈ X, . For a fixed v ∈ X F and for every w ∈ X F , the map τ w ,v belongs to E F . Thus, inequality (18) holds for any such τ w ,v and, summing over all those such maps, we obtain that, for every x ∈ X, Therefore, for every F ∈ F(G), v ∈ X F and x ∈ X, we have On the other hand, inequality (17) also yields that for every F ∈ F(G), w, v ∈ X F and x ∈ X, Then, from inequalities (19) and (20), we obtain that, for every F ∈ F(G), w, v ∈ X F and x ∈ X, (21) can be rewritten as Since µ is a DLR measure for φ, from Corollary 5.4 we obtain that µ(x)-almost surely it holds that Furthermore, notice that so that inequality (23) can be rewritten as For F ∈ F(G) and x ∈ X, define the following auxiliary probability measure over X F : Thus, inequality (24) yields that µ(x)-almost surely it holds that . Now, given y ∈ X, notice that the tail configuration x F c can be replaced by y F c with a penalty of 2∆ F (φ) as follows π y F (w) exp(−2∆ F (φ)) ≤ π x F (w) ≤ π y F (w) exp(2∆ F (φ)), so that (25) exp Moreover, it is easy to verify that Therefore, for every w ∈ X F , y ∈ X, it holds µ(x)-almost surely that and that Thus, concluding the proof.
We now state the main theorem of this subsection.
Theorem 5.8. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . If µ is a DLR measure for φ, then, for every > 0, there exist K ∈ F(G) and δ > 0 such that for every (K, δ)-invariant set F and x ∈ X, it holds µ(x)-almost surely that In particular, µ is a Bowen-Gibbs measure for φ.
Proof. Indeed, for every > 0, we obtain, from Proposition 2.2, Lemma 2.3, and Theorem 3.8, that there exist K ∈ F(G) and δ > 0 such that, for every (K, δ)-invariant set F ∈ F(G), respectively. Considering a sufficiently large K and sufficiently small δ so that the three conditions are satisfied at the same time, we obtain from Proposition 5.7 that Integrating this inequality with respect to dµ(x), it follows that µ is a Bowen-Gibbs measure for φ.

Existence of conformal measures.
In order to guarantee that the equivalences we prove here are non-trivial, we prove the existence of a conformal measure for an exp-summable potential with summable variation in the context of a countably infinite state space over an amenable group. The strategy is to apply a version of Prokhorov's Theorem.

Definition 11.
A sequence of probability measures {µ n } n in M(X) is tight if for every > 0 there exists a compact set K ⊆ X such that We now state a version of Prokhorov's Theorem (see [8,47]).
Theorem 5.9. Every tight sequence of probability measures in M(X) has a weak convergent subsequence.
Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Consider A ⊆ N a finite alphabet. Then φ| A G is also an exp-summable potential with summable variation according to {E m } m and the specification defined by equation (7) is quasilocal. Moreover, the set of Borel probability measures on A G is compact. Then, following [30,Comment (4.18)], for all x ∈ A G , any accumulation point of the sequence {γ Em (·, x)} m , will be a DLR measure µ. Finally, if we want to obtain a G-invariant DLR measure, for each g ∈ G, let gµ be given by gµ(A) = µ(g −1 · A), for any A ∈ B. Notice that, for every g ∈ G, the measure gµ is also a DLR measure for φ| A G due to the G-invariance of γ (see Corollary 4.10). Then it suffices to consider any accumulation point of the sequence 1 |Fn| g∈Fn gµ n , for a Følner sequence {F n } n . Now, let {A n } n in F(N) be a fixed exhaustion of N and, for each n ∈ N, denote the set of DLR measures and G-invariant DLR measures for φ n = φ| A G n by G n (φ) and G I n (φ), respectively. For each n ∈ N and each µ n ∈ G I n (φ), consider its extensionμ n ∈ M(X) given bỹ µ n (·) = µ n (· ∩ A G n ). The next result establishes that {μ n } n is tight and the reader can compare this to [45,Lemma 5.15].
Lemma 5.10. Let φ : X → R be an exp-summable potential with summable variation according to some exhausting sequence {E m } m . Then, for any sequence {µ n } n with µ n ∈ G I n (φ), for all n ∈ N, the sequence of extensions {μ n } n is tight.
Proof. Fix some n ∈ N. Then, for any a ∈ N and any y ∈ A {1 G } c n , Proposition 5.7 yields that If a / ∈ A n , thenμ n ([a]) = 0. On the other hand, if a ∈ A n , then for every y ∈ A where C(φ) = 2V E1 (φ) + 3δ(φ) Now, let > 0. Since φ is exp-summable, for each m ∈ N, there must exist a finite alphabet A ,m ∈ F(N) such that (26) b∈N\A ,m exp sup By Tychonoff's Theorem (see [47]), K is compact. Moreover, notice that where [a g ] = {x ∈ X : x(g) = a}. Therefore, for each n ∈ N, Since all the measures considered here are G-invariant, it follows that, for any y ∈ A where the fifth line follows from estimate (26). Therefore, for all n ∈ N,μ n (X \ K ) < , so that µ n (K ) = 1 −μ n (K c ) > 1 − , which proves the tightness of {μ n } n .
We have proven that for each sequence {µ n } n with µ n ∈ G I n (φ), the sequence {μ n } n of their extensions is tight. Then, the existence of at least one accumulation point is guaranteed by Prokhorov's Theorem. Let's see that an arbitrary accumulation point, which we will denote byμ, is conformal for φ and, moreover, that it is G-invariant.
Theorem 5.11. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m . Then, the set of G-invariant DLR measures for φ is non-empty.
Proof. Let {µ n } n be such that, for each n ∈ N, µ n is a G-invariant conformal measure for φ n : A G n → R (or, equivalently, µ n is a G-invariant DLR measure for φ n ). Thus, for each n ∈ N, any K ∈ F(G), and any τ ∈ E K,An , Indeed, let ψ : X → R be a bounded continuous potential. Observe that, for τ ∈ E K,An , (φ n ) τn 1 (B)). Then, we obtain where ψ n = ψ| A G n . Furthermore, Lemma 5.10 guarantees that the sequence of induced measures {μ n } n is tight and we can apply Prokhorov's Theorem to guarantee the existence of a limit point for some subsequence {µ n k } k , which we denote byμ. Now, we are going to prove thatμ is a conformal measure for φ. For that, consider a bounded continuous potential ψ : X → R, A ∈ F(N), K ∈ F(G), and τ ∈ E K,A . Then, where the fourth equality follows from the fact that for k large enough, A ⊆ A n k , and the last equality follows from weak convergence and the fact that ψ exp φ τ * is a continuous and bounded function. Indeed, first notice that φ τ * is a uniform limit of continuous functions that are bounded from above, since φ is exp-summable. Therefore, the same holds for φ τ * , so that exp(φ τ * ) is continuous and bounded (from above and below). Since A, K, and τ are arbitrary, this proves thatμ is conformal for φ and, therefore, DLR for φ.
It remains to show thatμ is G-invariant. For that, notice that, due to the weak convergence, for any B ∈ B,μ (g · B) = lim where we have used that, for each k ∈ N,μ n k is G-invariant due to G-invariance of A G n and to the fact that µ n k is G-invariant.

Finite entropy Bowen-Gibbs measures are equilibrium measures.
Thus far, we have proven that if φ : X → R is an exp-summable potential with summable variation, then a measure µ ∈ M(X) is a DLR measure if and only if it is a conformal measure. Also, if µ is a DLR measure, then µ is also a Bowen-Gibbs measure. For Bowen-Gibbs measures, we begin by exploring some equivalent hypothesis to having H F (µ) < ∞ for every F ∈ F(G), or, equivalently, to have finite Shannon entropy at the identity element. This will allow us to assume, indistinctly, that the energy of the potential is finite. The following lemma generalizes [43,Lemma 3.4]. Proof. Begin by noticing that, since µ is a Bowen-Gibbs measure for φ, we have that, in particular, for = 1, there exist K ∈ F(G), δ > 0, and a (K, δ)-invariant set F ∈ F(G) with 1 G ∈ F such that, for every x ∈ X, it holds that (28) exp We now prove that i) =⇒ iii) =⇒ ii) =⇒ i).
[i) =⇒ iii)] Notice that, since φ has summable variation according to {E m } m , then, in particular, φ has finite oscillation. Therefore, the result follows directly from Proposition 5.2, disregarding whether µ is a Bowen-Gibbs measure for φ or not.
[iii) =⇒ ii)] Begin by noticing that, due to standard properties of Shannon entropy, H(µ) ≤ H F (µ) ≤ |F |H(µ). Then, Thus, Now, due to exp-summability, without loss of generality we can assume that φ(x) ≤ 0, for all x ∈ X, Moreover, notice that if m = |F | − 1 and g 1 , · · · , g m is an enumeration of F \ {1 G }, then Notice that, due to the same argument as in the proof of [iii) =⇒ ii)], we have that Therefore, since exp (−|F |(1 + p(φ)) Z 1 G (φ) |F |−1 > 0, it suffices to prove that We now proceed to prove that Bowen-Gibbs measures with finite Shannon entropy at the identity are equilibrium measures. Proof. Since µ is a Bowen-Gibbs measure for φ, for every > 0, there exist K ∈ F(G) and δ > 0, such that for every (K, δ)-invariant set F ∈ F(G) and x ∈ X, Moreover, notice that, for every x ∈ X and F ∈ F(G), Therefore, where the second line follows from the G-invariance of µ and the third line follows from Lemma 2.3. On the other hand, after taking logarithm in equation (29) and dividing by |F |, we obtain Thus, for every x ∈ X and every (K, δ)-invariant set F ∈ F(G), Integrating the last equation with respect to µ, we get Therefore, if we take limit as F becomes more and more invariant, we have that where the last inequality follows from inequality (30). Since > 0 is arbitrary, we obtain that The reverse inequality follows from Proposition 5.3 and this concludes the proof.
5.6. Equilibrium measures are DLR measures. In §5.4, we proved that if φ : X → R is an exp-summable potential with summable variation according to an exhausting sequence {E m } m , then the set of G-invariant DLR measures for φ is non-empty. Throughout this section, fix a G-invariant ν ∈ G(φ). Given E ∈ F(G) and µ ∈ M G (X), denote by f µ,E the Radon-Nikodym derivative of µ| E with respect to ν| E , where µ| E and ν| E denote the restrictions of µ and ν to B E , respectively. More precisely, for every x ∈ X, .
Notice that f µ,E is well-defined, because any DLR measure for φ, in our context, is fully supported. Moreover, we can understand it as the pointwise limit of the simple functions f n µ, , where {A n } n is a fixed exhausting sequence of finite alphabets.
Consider the function ψ : Define, for each n ∈ N and E ∈ F(G), the simple function I n µ, , so we can define a measurable function I µ,E by considering the pointwise limit When there is no ambiguity, we will omit the subscript µ from the previous notations. Observe that, by the Monotone Convergence Theorem, In addition, so thatˆI We define the relative entropy of a measure µ ∈ M G (X) with respect to ν to be since the supports of f n E and f n F are contained in A G n and B E is generated by cylinder sets of this form. If v / ∈ A E n , then both sides of equation (32) are 0 and the result is proven. Otherwise, if v ∈ A E n , then Thus, Proof. Let E ∈ F(G). Since ν is a DLR measure for φ, by Theorem 5.8, ν is a Bowen-Gibbs measure for φ. Then, for every > 0, there exist K ∈ F(G) and δ > 0 such that for all (K, δ)-invariant set F ∈ F(G), the following conditions hold at the same time: Observe that, by considering the lower bound of the equation above, for any (K, δ)-invariant set F . Then, we have that ). First, observe that for any E, we can find a (K, δ)-invariant set F such that E ⊆ F . Then, by Lemma 5.14, Finally, by considering the upper bound given by the definition of Bowen-Gibbs measure and using a similar argument, we obtain that Since was arbitrary, we conclude that In particular, given φ : X → R an exp-summable potential with summable variation according to an exhausting sequence {E m } m , a G-invariant measure µ is an equilibrium measure for φ if and only if h(µ | ν) = 0, for some (or every) DLR measure ν. The next proposition is a generalization of Step 1 in the proof of [30,Theorem 15.37].
Proposition 5.16. Let φ : X → R be an exp-summable potential with summable variation according to an exhausting sequence {E m } m and µ ∈ M G (X) be an equilibrium measure for φ. Then, for every α > 0 and K ∈ F(G), there exists E ∈ F(G) such that K ⊆ E and Proof. Pick δ > 0 small enough so that every (K, δ)-invariant set F ∈ F(G) satisfies Int K (F ) = ∅. Consider 0 < < 1 and a tiling T with (K, δ)-invariant shapes, which we can do by Theorem 3.4.
Then, from Lemma 3.5, for every (S T , )-invariant set F ∈ F(G), there exist center sets C F (S) ⊆ C(S) ∈ C(T ) for S ∈ S(T ) such that F ⊇

S∈S(T )
Consider α as in Lemma 5.17, that is, whenever E ⊇ B and H E (µ|ν) − H E\B (µ|ν) ≤ α, then ν f E − f E\B ≤ . Now, using Proposition 5.16, fix a set E ∈ F(G) such that E ⊇ B and H E (µ|ν)− H E\B (µ|ν) ≤ α. Therefore, by the monotonicity of the relative entropy, we obtain that We now compute |µγ K (h) − µ(h)|. First observe that sinceh is B B\K -measurable and B ⊆ E, theñ h is B E\K -measurable. Therefore, recalling that µγ K (h) = µ(γ K h), We begin by justifying the terms that vanished from the first inequality to the second. Notice that )) and, in addition, since ν is a DLR measure, we have that We now have to deal with the three other terms. Notice that Since > 0 and h : X → R are arbitrary, we obtain that, µγ K = µ, which concludes the result.

Final considerations
In this section we consider the case when the group is finitely generated, which includes the wellstudied case G = Z d and show that our approach generalizes previous ones. Next, we present a version of Dobrushin's Uniqueness Theorem adapted to our framework and we apply it to a concrete class of examples of potentials defined in the G-full shift for any countable amenable group G.
6.1. The finitely generated case. We now restrict ourselves to the case that G is a finitely generated group. The main goal is to prove that our definition of a Bowen-Gibbs measure (Definition 9) for a given exp-summable potential with summable variation according to an exhausting sequence is related to the standard -but more restrictive -way to define Bowen-Gibbs measures (e.g., [45,38]). For that, we will prove that the bounds in Definition 9 can be replaced by a bound which involves the size of the boundary of invariant sets.
Suppose that G is finitely generated and let S be a finite and symmetric generating set. Without loss of generality, suppose that 1 G ∈ S. In this context, it is common to implicitly consider an exhausting sequence E m+1 = S m . For example, if G = Z d and S is the set of all elements s ∈ Z d with s ∞ ≤ 1, the sequence {E m } m recovers the notion of "boxes" with sides of length 2m + 1 centered at the origin, which is the most usual in the literature. In particular, one recovers the more standard definition of summable variation for a potential φ : X → R, which is given by where B(1 G , m) = S m denotes the ball of radius m (according to the word metric), ∂F := SF \ F denotes the "(exterior) boundary" of a set F , and |∂B(1 G , m)| is proportional to m d−1 in the Z d case.
Usually, potentials that have summable variation according to this particular exhausting sequence are called regular (see, for example, [38]). Notice that when {E m } m is an exhausting sequence of the form S m , we have that where ∂ int F = ∂F c denotes the "interior boundary" of F . Indeed, if g ∈ S m+1 F \ S m F , there must exist h ∈ ∂ int F such that d S (g, h) = m + 1, where d S denotes the word metric. In addition, we also have that |∂ int F | ≤ |S||∂F |, so From this, it is direct that On the other hand, if x, y ∈ X are such that x F = y F , we have that Therefore, we conclude that ∆ F (φ) ≤ V (φ)|S| 2 |∂F |. We now provide an alternative way of proving Proposition 2.2 and Lemma 2.3. Begin by noticing that a finitely generated group is amenable if and only if lim F →G |∂F | |F | = 0 (indeed, given > 0, we have that |∂F | ≤ |SF F | < · |F | for every (S, )-invariant set F ). Therefore, if φ has summable variation, it follows that and, similarly, In particular, in this context, we could alternatively have defined a Bowen-Gibbs measure as follows: if G is a finitely generated amenable group with generating set S and φ : X → R is an exp-summable potential with summable variation according to {S m } m , a measure µ ∈ M(X) is a Bowen-Gibbs measure for φ if for every > 0, there exist K ∈ F(G) and δ > 0 such that for every (K, δ)-invariant set F ∈ F(G) and x ∈ X, where C > 0 is a constant that we can choose to be C := 5V (φ)|S| 2 ≥ 2V (φ)|S| + 3∆(φ)|S| 2 .
This recovers the more standard definition of Bowen-Gibbs measure in terms of boundaries. Furthermore, with this choice of C, it is not difficult to check that we could mimic the proofs of Proposition 5.7, Theorem 5.8, and Theorem 5.13, thus providing all the implications involving Bowen-Gibbs measures. 6.2. Dobrushin's Uniqueness Theorem. From §5.4, we know that if φ : X → R is an exp-summable potential with summable variation according to an exhausting sequence {E m } m , then the set of Ginvariant DLR measures for φ is non-empty. One natural question that may arise is under which conditions we have uniqueness of the DLR measure. When a specification is a Gibbsian specification, the Dobrushin's Uniqueness Theorem (see [30]) addresses this question. For a detailed proof of a version of this theorem adapted to our setting, see [11].
Let 2 N be the set of all subsets of N, which is a σ-algebra, and M(N, 2 N ) be the set of probability measures on (N, 2 N ). For A ∈ 2 N , w ∈ X, and g ∈ G, denote where γ is a specification, notice that, for each x ∈ X, γ 0 g (·, x) ∈ M(N, 2 N ). Now, for each h ∈ G, the w h -dependence of γ 0 {g} (·, w) is estimated by the quantity where, for any given µ,μ ∈ M(N, 2 N ), µ −μ = max A∈E |µ(A) −μ(A)| (see [30, §8.1]). The infinite matrix ρ(γ) = (ρ gh (γ)) g,h∈G is called Dobrushin's interdependence matrix for γ. When there is no ambiguity, we will omit the parameter γ from the notation.

Theorem 6.1 (Dobrushin's Uniqueness Theorem).
If γ is a specification that satisfies the Dobrushin's condition, then there is at most one measure that is admitted by the specification γ.
We now present an example of a potential inspired by the Potts model [29,27] such that, under some conditions to be presented, is exp-summable and has summable variation according to an exhausting sequence {E m } m . Moreover, this potential will also satisfy that, if µ is a Bowen-Gibbs measure, φdµ > −∞. Another important property of this potential is that it is non-trivial, in the sense that it depends on every coordinate of G. We will also explore conditions on β > 0 such that the potential βφ satisfies Dobrushin's condition.
Later, choosing a n 3 ∈ N, great enough Therefore, the potential βφ is exp-summable, for all β > 0.
Remark 9. The set of functions c : G × N → [0, ∞) satisfying conditions (1) and (2) is non-vacuous. For example, given an exhausting sequence {E m } m , consider c : G × N → [0, ∞) and some constant L ≥ 0 such that (a) for every m ≥ 1, 0 ≤ c(g, n) ≤ L2 −m−1 |Em+1| 2 for every g ∈ E m+1 \ E m ; and (b) any c(1 G , n) of polynomial order will satisfy condition (2). We now prove that any such c satisfies the previous conditions (1) and (2). Due to (a) and the fact that {E m } m is nested, for every m ≥ 1, Then, so condition (1) is satisfied. Now, due to (b), we have that condition (2) is satisfied.
Our next goal is to study under which conditions we have uniqueness of Gibbs measures for βφ, where β can be interpreted as the inverse of the temperature of the system. For that, we use the Dobrushin's Uniqueness Theorem (Theorem 6.1). In order to obtain explicit conditions on β, we divide the rational into claims. Claim 1. If x, y ∈ X are such that x G\{g} = y G\{g} , for some g ∈ G, then h∈G (φ(h · x) − φ(h · y)) converges absolutely. Moreover, h∈G (φ(h · x) − φ(h · y)) = − c(1 G , x(g)) + c(1 G , y(g)) Proof of Claim 1. Since we are summing over all h-translations of x and y, for h ∈ G, we can assume, without loss of generality, that g = 1 G , that is, x G\{1 G } = y G\{1 G } . Then,  + c(g, y(1 G ) Note that if h = 1 G and g = h −1 , we have that x(h) = y(h) and x(gh) = y(gh), so that x(gh) = ≤ c(1 G , x(1 G )) + c(1 G , y(1 G )) + g∈G g =1 G c(g, x(1 G )) + c(g, y(1 G )) + c(g −1 , x(g)) . Therefore, from where it follows that, due to condition (1), h∈G |φ(h · x) − φ(h · y)| < ∞.
Moreover, notice that, applying the same rational with no absolute values, we get that h∈G (φ(h · x) − φ(h · y)) = −c(1 G , x(1 G )) + c(1 G , y(1 G )) + where γ βφ is the specification given by equation (7) for the potential βφ. Thus, if Dobrushin's condition is satisfied and, by Theorem 6.1, we have at most one DLR measure for the potential βφ. Furthermore, if β > 0, then the set of G-invariant DLR measures for βφ is non-empty, so that we can guarantee that if β ∈ 0, 1 4 h =1 G C(h) , there exists exactly one DLR measure for βφ.