2.1 Countable amenable groups

Let G be a countable group and denote the identity of G by e. We use the notation $K \Subset G$ to indicate that K is a finite subset of G. A Følner sequence for G is a collection of finite subsets $\{ F_n \}$ of G such that $G = \bigcup _{n \in \mathbb {N}} F_n$ and for all $K \Subset G$ , $\lim _{n \rightarrow \infty } |K F_n \triangle F_n| / |F_n| = 0$ . A countable group G is called amenable if there exists a Følner sequence in G.

For a given Følner sequence $\{ F_n \}$ , we say the sequence is tempered if there exists some $C> 0$ such that for all $n> 0$ , $| \bigcup _{k < n} F_k^{-1} F_n | \leq C |F_n |.$ For any Følner sequence, there exists a subsequence that is tempered, see [Reference Lindenstrauss16, Proposition 1.5] for a proof of this fact. A deeper discussion of Følner sequences and their relevant properties can be found in [Reference Ceccherini-Silberstein, Coornaert, Ceccherini-Silberstein and Coornaert8, Ch. 4], but for our purposes, we note that for any amenable group G, there exists a tempered Følner sequence that can be taken such that $n!$ divides $|F_n|$ for every $n \in \mathbb {N}$ .

Similar in spirit, we can define a relative almost-invariance: for any finite $F, T \Subset G$ and $\epsilon> 0$ , we say T is right $(K, \epsilon )$ -invariant if $|TK \triangle T| / |T| < \epsilon $ . We can equivalently say G is amenable if for every finite $K \Subset G$ and every $\epsilon> 0$ , there exists some finite $T \Subset G$ that is right $(K, \epsilon )$ -invariant.

In our setting, we will also be interested in a sense of sparseness of sets, for this, we define the following.

Definition 2.1. For any $S \subset G$ , $F \Subset G$ , we say S is left F-sparse if for all distinct $s, s' \in S$ ,

$$ \begin{align*} sF \cap s' F = \emptyset.\end{align*} $$

Note here, by a trivial application of the definition, we know for any $F \subset H \Subset G$ and $S \subset G$ , if S is left H sparse, then S is left F sparse. We will now define the right (respectively left) H-interior and H-boundary of F for $F, H \Subset G$ .

Definition 2.2. The right H-interior of F, denoted by $\mathrm {Int}_H(F)$ , is defined by

$$ \begin{align*}\mathrm{Int}_H(F) = \bigcap_{h \in H} Fh = \{ f \in F : \text{for all } h \in H, fh \in F \}.\end{align*} $$

The right H-boundary of F, denoted $\partial _H(F)$ , is all elements of F not in the H-interior. Precisely,

$$ \begin{align*}\partial_H(F) = F \backslash \mathrm{Int}_H(F) = \{ f \in F : \text{there exists } h \in H \text{ such that } fh \notin F \}.\end{align*} $$

The left H-interior/boundary of F is defined similarly.

The proof of our main theorem will also require taking advantage of a result by Downarowicz, Huczek, and Zhang regarding exact tilings of G from [Reference Downarowicz, Huczek and Zhang10]. For a countable amenable group G, a finite tiling $\mathcal {T} = \{ T_i : 1 \leq i \leq k \}$ is a collection of tiles such that $\bigcup _{i \leq k} T_i = G$ and for each $T_i \in \mathcal {T}$ , there exists a finite shape $S_i \Subset G$ , and a collection of centers $C(S_i) \subset G$ such that $T_i = S_i C(S_i)$ . In particular, a tiling $\mathcal {T}$ is called exact if for distinct $c_1 , c_2 \in C(S_i)$ , $S_i c_1 \cap S_i c_2 = \emptyset $ . We also note it can be assumed that $e \in S$ for every shape. We can now state their result.

Their result in fact included statements regarding the entropy of the tiling space which we have omitted since they are not necessary for the purposes of this paper.

2.2 G-subshifts

Let G be a countable amenable group. Let $\mathcal {A}$ be a finite set, called the alphabet, endowed with the discrete topology. Our configuration space is the set of functions $x: G \rightarrow \mathcal {A}$ , which we denote $\mathcal {A}^G$ , endowed with the product topology. For any $H \subset G$ and $x \in \mathcal {A}^G$ , we denote the restriction of x to H by $x_H$ .

We define a G-action of homeomorphisms on $\mathcal {A}^G$ by the left-translation map: for all $g \in G$ and $x \in \mathcal {A}^G$ , we define pointwise $\sigma _g(x)_h = x_{gh}$ . The set $\mathcal {A}^G$ and the collection of left-translations $(\sigma _g)_{g \in G}$ together form a topological dynamical system $(\mathcal {A}^G, \sigma )$ which we call the full G-shift on $\mathcal {A}$ . A G-subshift is a subset $X \subset \mathcal {A}^G$ that is closed in the product topology and is $\sigma $ -invariant (that is, for all $g \in G$ , $\sigma _g(X) \subset X$ ). When G is clear, we will refer to a G-subshift as just a subshift.

For any finite $F \Subset G$ and any $w \in \mathcal {A}^F$ , we call w a configuration of shape F. For a subshift X and a finite configuration w of shape $F \Subset G$ , we say w is in the language of X (or w is legal in X) if there exists some $x \in X$ such that $x_F = w$ . We call $L_F(X) = \{ x_F : x \in X \}$ the F-language of X, and $L(X) = \bigcup _{F \Subset G} L_F(X)$ the language of X.

For any $F \Subset G$ and $w \in \mathcal {A}^F$ , we define the cylinder set of w as follows:

$$ \begin{align*}[w] = \{ x \in \mathcal{A}^G : x_F = w \}.\end{align*} $$

In particular, the cylinder sets form a basis for the product topology on $\mathcal {A}^G$ . Cylinder sets are frequently taken intersected with a subshift, which will be clear by context.

Finally, we note here that the product topology on $\mathcal {A}^G$ is metrizable, and for any Følner sequence $\{ F_n \}$ , the metric

$$ \begin{align*}d(x, y) = 2^{ - \min \{ n \in \mathbb{N} : x_{F_n} = y_{F_n} \} }\end{align*} $$

induces the product topology. This metric serves primarily to establish that G-subshifts are expansive.

For any disjoint $H, K \subset G$ and any $x \in \mathcal {A}^H$ , $y \in \mathcal {A}^K$ , we define the concatenation of x and y by $xy \in \mathcal {A}^{H \cup K}$ such that $(xy)_H = x$ and $(xy)_K = y$ . We also define for any $v \in L_F(X)$ , the extender set of v, $E_X(v)$ , to be the collection of all background configurations for which v is legal. Specifically, $E_X(v) = \{ \eta \in \mathcal {A}^{G \backslash F} : v\eta \in X \}$ . For any $v, w \in L_F(X)$ , we say v is replaceable by w if $E_X(v) \subset E_X(w)$ .

We will now define the following functions that will be useful in the proof of our main results and essential to our discussion of the conformal Gibbs corollaries.

Note this is exactly switching v and w in the F-location whenever the resulting point is still in X. It will be noted in §4 that these functions are in fact Borel isomorphisms that generate the homoclinic (Gibbs) relation.

For a given subshift $X \subset \mathcal {A}^G$ , we will denote the collection of $\sigma $ -invariant probability measures on X by $\mathcal {M}_\sigma (X)$ . The existence of such measures is guaranteed by the fact that G is amenable. For a given $\mu \in \mathcal {M}_\sigma (X)$ , for any collection of legal finite configurations $W \subset L(X)$ , we will use $\mu (W) $ to mean $\mu ( \bigcup _{w \in W} [w] )$ .

We say $\mu \in \mathcal {M}_\sigma (X)$ is ergodic if for all measurable $A \subset X$ , for all $g \in G$ , if $\mu ( A \triangle \sigma _g^{-1} A) = 0$ , then $\mu (A) \in \{ 0, 1\}$ . Here, $\mathcal {M}_\sigma (X)$ is convex and compact under the weak- $*$ topology, in fact, it forms a Choquet simplex whose extreme points are exactly the ergodic measures.

2.3 Thermodynamic formalism

The theory of thermodynamic formalism bridges the gap between microscopic and macroscopic descriptions of systems with many interacting particles, extending concepts of statistical mechanics to symbolic dynamics. Gibbs measures are central to this framework, enabling the analysis of global statistical properties derived from local interactions in a broad range of dynamical systems. This section delves into the thermodynamic formalism for countable amenable group actions on finite alphabet subshifts, examining topological pressure and its connection to statistical physics, the construction of partition functions, and the characterization of equilibrium states.

To begin, we must define topological pressure of a given potential over our subshift X. We first let $\phi : X \rightarrow \mathbb {R}$ be a real valued, continuous function which we will call a potential. We will denote the set of all potentials on a subshift X by $C(X)$ . For any $F \Subset G$ , we define the F-Birkhoff sum of $\phi $ : $\phi _F = \sum _{g \in F} \phi \circ \sigma _g$ . Finally, for any open cover $\mathcal {U}$ of X, any $F \Subset G$ , we define the open cover $\mathcal {U}^F = \bigvee _{f \in F} \sigma _f^{-1} \mathcal {U} = \{ \bigcap _{f \in F} \sigma _f^{-1}(U_f) : U_f \in \mathcal {U} \}$ . We use this notation to define the following partition function.

Definition 2.5. We define a partition function for any $F \Subset G$ and any open cover $\mathcal {U}$ of X:

$$ \begin{align*}Z_F(\phi, \mathcal{U}) = \inf \bigg\{\! \sum_{u \in \mathcal{U}'} \mathrm{exp }\big( \sup_{x \in u} \phi_F(x) \big) : \mathcal{U}' \text{ is a subcover of } \mathcal{U}^F \bigg\} .\end{align*} $$

We can define the topological pressure of $\phi $ with respect to a given open cover $\mathcal {U}$ to be

$$ \begin{align*}P(\phi, \mathcal{U}) = \lim_{n \rightarrow \infty} |F_n|^{-1} \log Z_{F_n}(\phi, \mathcal{U})\end{align*} $$

for any Følner sequence $\{ F_n \}$ . This limit is guaranteed to exist and does not depend on the choice of Følner sequence, further discussion of which can be found in [Reference Bufetov5] and [Reference Gurevich and Tempelman12]. We can now define the following.

Definition 2.6. The topological pressure of $\phi $ is then

$$ \begin{align*}P_{\mathrm{top}}(\phi) = \sup_{\mathcal{U}} P(\phi, \mathcal{U}).\end{align*} $$

Since, in the subshift setting, the dynamical system $(X, \sigma )$ is expansive, the above supremum is attained at an open cover $\mathcal {U}$ of diameter less than or equal to the expansiveness constant. In particular, we can compute topological pressure by

$$ \begin{align*} P_{\mathrm{top}}(\phi) = \lim_{n \rightarrow \infty} |F_n|^{-1} \log \sum_{w \in L_{F_n}(X)} \mathrm{exp }\big( \sup_{x \in [w]} \phi_{F_n}(x) \big). \end{align*} $$

In addition to the topological pressure, for a given invariant measure $\mu \in \mathcal {M}_\sigma (X)$ , we can define the pressure of $\mu $ with respect to $\phi \in C(X)$ as follows:

$$ \begin{align*} P_\phi(\mu) = h(\mu) +\int \phi \,d\mu, \end{align*} $$

where $h(\mu )$ is the Kolmogorov–Sinai entropy of the invariant probability measure $\mu $ . As shown by Ollagnier and Pinchon [Reference Ollagnier and Pinchon19], in the countable amenable subshift setting, the variational principle holds. In particular, for all $\phi \in C(X)$ ,

$$ \begin{align*}P_{\mathrm{top}}(\phi) = \sup_{\mu \in \mathcal{M}_\sigma(X)} P_\phi(\mu) .\end{align*} $$

In statistical physics, equilibrium states correspond with probability measures on the state space that minimize the Gibbs free energy of the system. Up to a multiplicative constant, the free energy of an invariant measure corresponds with the negative pressure, and so we similarly define an equilibrium state as follows.

Definition 2.7. We say $\mu \in M_\sigma $ is an equilibrium state for $\phi $ if it attains the variational principle supremum, that is,

$$ \begin{align*}P_{\mathrm{top}}(\phi) = h(\mu) + \int \phi \,d\mu .\end{align*} $$

When h is upper semicontinuous (as in the case for expansive dynamical systems), we know the collection of equilibrium states for a given $\phi $ is non-empty, compact under the weak- $*$ topology, and convex, where the extreme points are exactly the ergodic equilibrium states.

2.4 Equilibrium measures and the Gibbs property

A rich theory has developed around relating equilibrium states and measures with the Gibbs property (for some definition of Gibbs property). In the classical results of Dobrušin, Lanford, and Ruelle, the Gibbs property can be defined in terms of conditional probabilities as follows.

Definition 2.8. For a subshift $X \subset \mathcal {A}^{\mathbb {Z}^d}$ and a potential $\phi \in C(X)$ , we say a measure $\mu $ is Gibbs for $\phi $ if for all $F \Subset \mathbb {Z}^d$ , for all $w \in L_F(X)$ , and for $\mu $ -a.e. $\eta \in \mathcal {A}^{\mathbb {Z}^d \backslash F}$ such that $w \eta \in X$ ,

$$ \begin{align*}\mu( w || \eta ) = \frac{ \mathrm{exp }( \phi( w \eta )) }{\sum_{v \in L_F(X)} \mathrm{exp }( \phi( v \eta )) \cdot 1_X(v\eta)}.\end{align*} $$

In other words, we say a measure is Gibbs for $\phi $ when the probability of a local configuration, w, given a background configuration, $\eta $ , can be computed in the typical way one computes Gibbs measures in the finite case. The results of Dobrušin combined with those of Lanford and Ruelle show equivalence to the Gibbs property and being an equilibrium state under certain assumptions on subshift.

The following theorem involves a technical condition called property (D), which is described in [Reference Dobrushin9].

Subshifts of finite type, SFTs, are an important and well-studied class of subshifts that are not defined here. Analogous results have since been shown in the countable amenable subshift setting [Reference Ollagnier and Pinchon18, Reference Tempelman23]. Note that these statements rely on relatively strong assumptions on both the subshift and the potential. When no assumptions are made of the potential, little can be said about the equilibrium state. In fact, by upper semicontinuity of the entropy map, it can be shown that for any subshift $X \subset \mathcal {A}^G$ and any ergodic state $\mu \in \mathcal {M}_{\sigma }(X)$ , there exists a potential $\phi \in C(X)$ for which $\mu $ is the unique equilibrium state [Reference Jenkinson13]. It is therefore quite natural to retain some regularity assumptions on the class of potentials under consideration.

3.1 Sufficiently sparse almost partitions

The following lemma allows us to generate a finite $\epsilon $ -almost partition of G where each part is sufficiently sparse relative to some fixed $F \Subset G$ .

Lemma 3.1. For any $F \Subset G$ , $\epsilon> 0$ , and Følner sequence $\{ F_n \}$ , there exists a finite collection $\mathcal {P} = \{ P_i : 1 \leq i \leq N \}$ of pairwise disjoint, left F-sparse subsets of G and a subsequence $\{ F_{n_k} \}$ such that

$$ \begin{align*}\liminf_{k \rightarrow \infty} \frac{| \bigcup_{P \in \mathcal{P}}P \cap F_{n_k}|}{|F_{n_k}|} \geq 1-\epsilon .\end{align*} $$

In lieu of a complete and technical proof, we provide an outline of how this result can be shown. The result can be viewed as a weakened reformulation of a quasitiling result of Ornstein and Weiss in [Reference Ornstein and Weiss21], or an application of the exact tiling results in [Reference Downarowicz, Huczek and Zhang10]. In essence, one can construct a quasitiling (or an exact tiling) of G using a finite collection of shapes that are sufficiently F-invariant. The F-invariance ensures that we can consider only the F-interiors of these shapes and maintain a $(1-\epsilon )$ -covering of G. We then let $P \in \mathcal {P}$ represent all the shifts of a particular location of an element in a specified shape. This allows us to ensure that $\mathcal {P}$ is left F-sparse since we have restricted to the F-interiors of the relevant shapes.

3.2 Subshift lemmas

We begin by describing our regularity constraints imposed on the potential $\phi $ .

Definition 3.2. The F-variation of $\phi $ for any $F \subset G$ is defined as

$$ \begin{align*}\mathrm{Var}_{F}(\phi) = \sup \{ \phi(x) - \phi(y) : x, y \in X \text{ and } x_F = y_F \}. \end{align*} $$

Definition 3.4. We say $\phi $ has summable variation according to the exhaustive sequence $\{ E_n \}$ if

$$ \begin{align*}\sum_{n \in \mathbb{N} } |E_{n+1}^{-1} \backslash E_{n}^{-1} | \cdot \mathrm{Var}_{ E_n }(\phi) < \infty. \end{align*} $$

Additionally, we will say $\phi \in C(X)$ has summable variation if $\phi $ has summable variation according to some exhaustive sequence.

For a fixed exhaustive sequence $\{ E_n \}$ , we define $SV_{\{ E_n \}}(X)$ to be the collection of all potentials with summable variation according to $\{ E_n \}$ . We also let $SV(X) = \bigcup _{\{ E_n \}} SV_{\{ E_n \}}(X)$ denote the collection of all potentials with summable variation according to some exhaustive sequence.

It is worth noting here that when $G = \mathbb {Z}^d$ , we can take $E_n = \{ k \in \mathbb {Z}^d : |k_i| \leq n \}$ and summable variation according to this sequence corresponds to d-summable variation in the typical sense.

Definition 3.5. For an exhaustive sequence $\{ E_n \}$ , we define the summable variation norm on $SV_{\{ E_n \}}(X)$ by

$$ \begin{align*}|| \phi ||_{SV_{\{ E_n\}}} = 2 |E_1| \cdot || \phi||_\infty + \sum_{n \in \mathbb{N} } |E_{n+1}^{-1} \backslash E_{n}^{-1} | \cdot \mathrm{ Var}_{ E_n }(\phi). \end{align*} $$

Proof. First, let $H \Subset G$ and note $ HF^{-1}$ is also a finite set. We therefore have some $N \in \mathbb {N}$ such that for all $n \geq N$ , $ H F^{-1} \subset E_n$ and so $H \subset E_n F$ . It immediately follows that $\{ E_n F \}$ is an exhaustive sequence.

Since $e \in F$ by assumption, we have $E_n \subset E_n F$ . We now see that

$$ \begin{align*}\sum_{n \in \mathbb{N}} |E_{n+1} F \backslash E_n F| \mathrm{Var}_{E_nF}(\phi) \leq |F| \sum_{n \in \mathbb{N}} |E_{n+1} \backslash E_n | \mathrm{Var}_{E_n}(\phi) < \infty .\\[-45pt] \end{align*} $$

We now restate relevant definitions and lemmas from García-Ramos and Pavlov in [Reference García-Ramos and Pavlov11].

In particular, $O_v(u)$ represents the indices where v occurs in u and can be read as ‘occurrences of v in u’. We will now define a replacement function that for a given finite configuration u, replaces occurrences of v with w in specified locations.

Since S is an F-sparse subset of $O_v(u)$ , we know $u'$ is well defined and uniquely determined. Note here that $u'$ is exactly the configuration obtained by replacing v with w in the S-locations.

We remind the reader that for any finite configuration $v \in L_F(X)$ , the extender set of v is defined as $E_X(v) = \{ \eta \in \mathcal {A}^{F^c} : v \eta \in X \}$ . Note here, whenever $E_X(v) \subset E_X(w)$ , then $R_u^{v \rightarrow w}(S) \in L(X)$ for any left F-sparse set $S \subset O_v(u)$ .

Lemma 3.10. [Reference García-Ramos and Pavlov11, Lemma 4.3]

For any F and $v, w \in L_F(X)$ , any left F-sparse set $T \subset O_v(u)$ , any $u'$ , and any $m \leq |T \cap O_w(u')|$ ,

$$ \begin{align*}| \{ (u, S) : S \text{ is left } F\text{-sparse}, \; |S| = m, \; S \subseteq T, \; u' = R_u^{v\rightarrow w}(S) \} | \leq {|T \cap O_w(u')| \choose m}. \end{align*} $$

The following lemma will be useful for computing topological pressure.

Lemma 3.11. Let $\mu _\phi $ be an ergodic equilibrium measure for any $\phi \in C(X)$ . For any tempered Følner sequence $ \{ F_n \}$ , if $S_n \subset L_{F_n}(X)$ such that $\mu _\phi (S_n) \rightarrow 1$ , then

$$ \begin{align*}P_{\mathrm{top}}(\phi) = \lim_{n \rightarrow \infty} \frac{1}{|F_n|} \log \bigg( \sum_{w \in S_n} \sup_{x \in [w]} \mathrm{exp }(\phi_{F_n} (x) ) \bigg). \end{align*} $$

Proof. Let $\epsilon> 0$ and note by definition of topological pressure above, we have

$$ \begin{align*} \limsup_{n \rightarrow \infty} \frac{1}{|F_n|} \log \bigg( \sum_{w \in S_n} \sup_{x \in [w]} \mathrm{exp }(\phi_{F_n} (x) ) \bigg) \leq P_{\mathrm{top}}(\phi). \end{align*} $$

For every n, define

$$ \begin{align*} T_n = \Big\{ w \in L_{F_n}(X) : \mu_\phi([w]) < \mathrm{exp }\Big( |F_n| \Big(\sup_{x \in [w]} \phi_{F_n}(x) - (P_{\mathrm{top}}(\phi) - \epsilon) \Big) \Big) \Big\}. \end{align*} $$

Note since $F_n$ is a tempered Følner sequence and $\mu _\phi $ is ergodic, we can apply the pointwise ergodic theorem and the Shannon–Macmillan–Breiman theorem proven in [Reference Gurevich and Tempelman12, Reference Ornstein and Weiss21] to see for $\mu _\phi $ -a.e. $x \in X$ ,

$$ \begin{align*}P_{\mathrm{top}}(\phi) = h(\mu_\phi) + \int \phi \,d\mu_\phi = \lim_{n \rightarrow \infty} |F_n|^{-1} ( \phi_{F_n}(x) - \log \mu_\phi([x_{F_n} ]) ). \end{align*} $$

By the definition of $T_n$ , it therefore follows that $\mu _\phi ( \bigcup _{N \in \mathbb {N}} \bigcap _{n \geq N} T_n) = 1$ , and so $\mu _\phi (T_n) \rightarrow 1$ . Further, $\mu ( S_n \cap T_n) \rightarrow 1$ and by definition of $T_n$ ,

$$ \begin{align*}\sum_{w \in S_n \cap T_n} \sup_{x \in [w]} \mathrm{exp }( \phi_{F_n}(x) ) \geq \mu_\phi(S_n \cap T_n) \mathrm{exp }( |F_n|( P_{\mathrm{top}}(\phi) -\epsilon) ). \end{align*} $$

By taking sufficiently large n, we therefore have

$$ \begin{align*}\sum_{w \in S_n } \sup_{x \in [w]} \mathrm{exp }( \phi_{F_n}(x) ) \geq 0.5 \mathrm{exp }( |F_n|( P_{\mathrm{top}}(\phi) -\epsilon) ). \end{align*} $$

Since $\epsilon> 0$ was arbitrary, we are finished.

We remind the reader that for $v, w \in L_F(X)$ , $\xi _{v, w}: X \rightarrow X$ is the map that swaps v and w in the F location whenever legal.

Observation 3.12. If $\phi $ has summable variation according to the exhaustive sequence $\{ E_n \}$ , then for any $F \Subset G$ , $v, w \in L_F(X)$ , and any $x \in [v]$ ,

$$ \begin{align*} \sum_{g \in G} | \phi(\sigma_g(x)) - \phi(\sigma_g(\xi_{v, w}(x))) | \leq |F| \cdot ||\phi||_{SV_{\{ E_n\}}}. \end{align*} $$

We note the proof of Observation 3.12 is standard and the statement appears nearly identically as [Reference Barbieri, Gómez, Marcus, Meyerovitch and Taati1, Proposition 3.1]. The final lemma in this section is a Stirling approximation that will be necessary to compute a lower bound on pressure in the proof of Theorem 1.1.

Lemma 3.13. For $b, a \in \mathbb {Q}_+$ , and any sequence $n_k \in \mathbb {N}$ such that for all k, $k!$ divides $n_k$ , let $D \in \mathbb {R}$ satisfying $\log ({a}/{b})> D$ . Then, for sufficiently small $c \in \mathbb {Q}_+$ , we have

$$ \begin{align*} \lim_{k \rightarrow \infty} n_k^{-1} \bigg( \log { an_k \choose c n_k} - \log {b n_k \choose c n_k} \!\bigg)> cD .\end{align*} $$

Proof. We define for all $c \in (0, \min \{ a, b\} ) \cap \mathbb {Q}$ ,

$$ \begin{align*}f(c) = \lim_{k \rightarrow \infty} n_k^{-1} \bigg( \log { an_k \choose c n_k} - \log {b n_k \choose c n_k} \!\bigg) .\end{align*} $$

First note, we have for all $k \in \mathbb {N}$ ,

$$ \begin{align*}\bigg(\! \log { an_k \choose c n_k} - \log {b n_k \choose c n_k} \!\bigg) = \log (a n_k)! + \log ((b-c)n_k)! - \log (b n_k)! - \log ((a-c)n_k)!. \end{align*} $$

If we divide by $n_k$ and take the limit as k goes to infinity, Stirling’s approximation implies

$$ \begin{align*}f(c) = a \log \frac{a}{a-c} + b \log \frac{b-c}{b} + c \log \frac{a-c}{b-c} .\end{align*} $$

We now examine

$$ \begin{align*}f'(c) = \frac{a}{a-c} - \frac{b}{b-c} + \log \frac{a-c}{b-c} + c \frac{b-a}{(a-c)(c-b)}. \end{align*} $$

Note here $f'(0) = \log ({a}/{b})> D$ . Since $f(0) = 0$ , it immediately follows that for sufficiently small $c \in \mathbb {Q}_+$ , we know $f(c)> cD$ .

3.3 Proof of Theorem 1.1

Theorem 3.1. Let G be a countable amenable group and X be a G-subshift. Let $\phi \in SV(X)$ , $\mu _\phi $ an equilibrium state for $\phi $ , $F \Subset G$ and $v, w \in L_F(X)$ . If $E_X(v) \subset E_X(w)$ , then

$$ \begin{align*}\mu_\phi([v]) \leq \mu_\phi([w]) \cdot \sup_{x \in [v]} \mathrm{exp }\bigg( \sum_{g \in G} \phi(\sigma_g(x)) - \phi(\sigma_g(\xi_{v, w}(x))) \bigg).\end{align*} $$

First, we let X, $\phi $ , F, v and w be as in the statement of the theorem. By ergodic decomposition, it is sufficient to show the desired result for ergodic equilibrium states, so we let $\mu _\phi $ be an ergodic equilibrium state for $\phi $ . Fix $C = \sup _{x \in [v]} \sum _{g \in G} \phi (\sigma _g(x)) - \phi (\sigma _g(\xi _{v, w}(x)))$ and note by Observation 3.12, we know $-\infty < C < \infty $ . We now suppose for a contradiction that $\mu _\phi ([v])> \mu _\phi ([w]) \cdot e^C$ .

We now take $\delta \in (0, \frac {4 }{5} \mu _\phi ([v])) $ satisfying

$$ \begin{align*}e^{-C} \cdot \frac{\mu_\phi([v]) - 5 \delta / 4}{\mu_\phi([w]) + 2 \delta}> 1 .\end{align*} $$

Note here, this is attainable for all sufficiently small $\delta $ since we know

$$ \begin{align*}f(\delta ) = e^{-C} \cdot \frac{\mu_\phi([v]) - 5 \delta / 4}{\mu_\phi([w]) + 2 \delta} \end{align*} $$

is continuous and, by assumption, $f(0)> 1$ .

Let $F_n$ be a Følner sequence such that for each n, $n!$ divides $|F_n|$ and $F_n^{-1} = F_n$ . It is noted by Xu and Zheng in [Reference Xu and Zheng24] that we can assume $F_n$ is tempered by passing to a subsequence. Re-index this sequence by n, and thus since $F_n$ is tempered, we know it satisfies the requirements for the lemmas above.

We now define $S_n \subset L_{F_n}(X)$ as follows:

$$ \begin{align*}S_n := \{ u \in L_{F_n}(X) : |O_v(u)| \geq |F_n| (\mu_\phi([v]) - \delta) \text{ and }|O_w(u)| \leq |F_n| (\mu_\phi([w]) + \delta) \}. \end{align*} $$

We note here since $\mu _\phi $ is ergodic, $\mu _\phi ( S_n) \rightarrow 1$ and we can therefore apply Lemma 3.11 to see

$$ \begin{align*}P_{\mathrm{top}}(\phi) = \lim_{n \rightarrow \infty} |F_n|^{-1} \log \sum_{u \in S_n} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) ). \end{align*} $$

We now apply Lemma 3.1 with respect to F and $\epsilon = \delta / 8$ , re-index our Følner sequence according to the lemma, and let $\mathcal {P} = \{ P_i : 1 \leq i \leq N \}$ be our collection of pairwise disjoint, left F-sparse sets. We also pass to the tail of the sequence of Følner sets ensuring that for each n, we have

$$ \begin{align*} \frac{|\!\bigcup_{i \leq N} P_i \cap F_n|}{|F_n|} \geq 1 - \frac{\delta}{4}. \end{align*} $$

For each $1 \leq i \leq N$ , we now define

$$ \begin{align*}S_n^{i} = \bigg\{ u \in S_n : |O_v(u) \cap P_i| \geq \frac{\mu_\phi([v]) - 5 \delta / 4}{N} |F_n| \bigg\}. \end{align*} $$

Proof. Suppose there exists some $u \in S_n \backslash \bigcup _{1 \leq i \leq N} S_n^{i}$ . We therefore know for all $P \in \mathcal {P}$ , $|O_v(u) \cap P| < ({\mu _\phi ([v]) - 5 \delta / 4})/{N} |F_n| $ . Let $Q = F_n \backslash \bigcup _{i \leq N} P_i$ and note that

$$ \begin{align*}|O_v(u)| = \sum_{i=1}^N | O_v(u) \cap P_i| + |O_v(u) \cap Q| < (\mu_\phi([v]) - 5 \delta / 4) |F_n| + |O_v(u) \cap Q|. \end{align*} $$

However, since $u \in S_n$ , we know $(\mu _\phi ([v]) - \delta ) \cdot |F_n| < |O_v(u)|$ . It follows that

$$ \begin{align*}(\mu_\phi([v]) - \delta) \cdot |F_n| < (\mu_\phi([v]) - 5 \delta / 4) |F_n| + |O_v(u) \cap Q|. \end{align*} $$

By our construction of $\mathcal {P}$ and since we took a sufficiently long tail for $F_n$ , we know that $ |O_v(u) \cap Q| \leq |Q \cap F_n| \leq ({\delta }/{4})|F_n|$ . We therefore have $\mu _\phi ([v]) - \delta < \mu _\phi ([v]) -\delta $ , arriving at a contradiction, and we can conclude that $S_n = \bigcup _{1 \leq i \leq N} S_n^{i}$ .

Lemma 3.3. There exists some fixed $1 \leq i \leq N$ and a subsequence $(n_k)$ for which we have

$$ \begin{align*}P_{\mathrm{top}}(\phi) = \lim_{k \rightarrow \infty} |F_{n_k}|^{-1} \log \sum_{u \in S_{n_k}^i} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_{n_k}}(x) ). \end{align*} $$

Proof. First notice, since $S_n = \bigcup _{i \leq N} S_n^i$ ,

$$ \begin{align*}\sum_{ u \in S_n} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x)) \leq \sum_{i=1}^N \sum_{ u \in S_n^{i}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x)),\end{align*} $$

and we therefore have for each n, some $1 \leq i_n \leq N$ such that

$$ \begin{align*}\frac{1}{N} \sum_{u \in S_n} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x)) \leq \sum_{ u \in S_n^{i_n}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x)). \end{align*} $$

We now use the pigeonhole principle to find a subsequence $n_k$ such that $i_{n_k} = i$ is constant. Note, by our construction, we have

$$ \begin{align*}\sum_{u \in S_{n_k}^i} \sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x)) \geq N^{-1} \sum_{u \in S_n}\sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x)). \end{align*} $$

We now examine

$$ \begin{align*}\kern-15pt P_{\mathrm{top}}(\phi) &= \lim_{n \rightarrow \infty} |F_n|^{-1} \sum_{u \in S_n} \sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x))\\ &\geq \lim_{n \rightarrow \infty} |F_n|^{-1} \log \sum_{u \in S_{n_k}^i} \sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x))\\ & \geq \lim_{n \rightarrow \infty} |F_n|^{-1} \log N^{-1} \sum_{u \in S_n}\sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x))\\ &= \lim_{n \rightarrow \infty} |F_n|^{-1} \log N^{-1} + \lim_{n \rightarrow \infty} |F_n|^{-1} \sum_{u \in S_n} \sup_{x \in [u]} \mathrm{exp }(\phi_{F_n}(x)) = P_{\mathrm{top}}(\phi).\\[-47pt] \end{align*} $$

We now apply Lemma 3.3 and fix the resulting $1 \leq i \leq N$ and re-index along the resulting subsequence $(n_k)$ . Denote $S_k' = S_{n_k}^i$ and fix $P = P_i$ . We can therefore compute topological pressure by restricting to the sequence of collections of finite configurations $S_n^{\prime }$ . We will now make replacements of v with w in every $u \in S_n^{\prime }$ at locations in P with a small frequency to increase the pressure of the dynamical system to arrive at our contradiction.

We let $a, b \in \mathbb {Q}_+$ such that $a \leq (\mu _\phi ([v]) - 5\delta / 4) /N$ , $b \geq (\mu _\phi ([w]) + 2\delta ) / N $ , and $\log ({a}/{b})> C$ . Note, such $a, b \in \mathbb {Q}_+$ must exist since $\log ({ \mu _\phi ([v]) - 5\delta / 4})/ (\mu _\phi ([w]) + 2\delta )> C$ by assumption. We now let $\epsilon \in \mathbb {Q}_+$ be sufficiently small to satisfy Lemma 3.13 with respect to $a, b$ and $D = C$ .

Define for each $u \in S_n^{\prime }$ ,

$$ \begin{align*}A_u = \{ R_u^{v \rightarrow w}(S) : S \subset O_v(u) \cap P \text{ and } |S| = \epsilon \cdot |F_n| \}. \end{align*} $$

Since $\epsilon \in \mathbb {Q}$ and $n!$ divides $|F_n|$ , for sufficiently large n, $ \epsilon \cdot |F_n| \in \mathbb {N}$ and $A_u$ is well defined. We restrict our consideration for all n large enough that this definition makes sense. Additionally, since $E_X(v) \subset E_X(w)$ , we have $A_u \subset L(X)$ .

We can now define for each $n \in \mathbb {N}$ , $L_n = \bigcup _{u \in S_n^{\prime }} A_u$ .

Lemma 3.4. For each $n \in \mathbb {N}$ , $\sum _{u \in L_n} \sup _{x \in [u]} \mathrm {exp }( \phi _{F_n}(x) )$ is bounded below by

(1) $$ \begin{align} \frac{( \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) - ({ \epsilon \cdot |F_n| }/{ N}) C ) ) { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| }}{ { \lfloor (\mu_\phi([w]) + 2\delta) |F_n|/ N \rfloor \choose \epsilon \cdot |F_n| }}. \end{align} $$

Proof. Note here, for each $u \in S_n^{\prime }$ ,

$$ \begin{align*}|A_u| = {|O_v(u) \cap P| \choose \epsilon \cdot |F_n| } \geq { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| }. \end{align*} $$

However, for every $u' \in \bigcup _{u \in S_n} A_u$ , we know $u'$ came from some $u \in S_n$ by replacing the $\epsilon \cdot |F_n| / N$ v terms with w. We can therefore get a bound on the following:

$$ \begin{align*}|O_w(u') \cap P| \leq |O_w(u) \cap P| + \frac{\epsilon \cdot |F_n|}{N} < \frac{(\mu_\phi([w]) + \delta) |F_n| }{N} + \frac{\delta |F_n|}{|FF^{-1}| \cdot N}. \end{align*} $$

Since $|FF^{-1}| \geq 1$ , we therefore have

$$ \begin{align*}|O_w(u') \cap P| < \frac{(\mu_\phi([w]) + 2\delta) |F_n|}{N}. \end{align*} $$

We therefore know for any fixed $u' \in A_u$ , there are at most

$$ \begin{align*}{ \lfloor (\mu_\phi([w]) + 2\delta) |F_n| /N \rfloor \choose \epsilon \cdot |F_n| }\end{align*} $$

u terms for which $u' \in A_u$ .

It follows that $\sum _{u \in L_n} \sup _{x \in [u]} \mathrm {exp }( \phi _{F_n}(x) )$ is bounded below by

(2) $$ \begin{align} \frac{( \sum_{u \in S_n^{\prime}} \sum_{u' \in A_u} \sup_{x \in [u']} \mathrm{exp }( \phi_{F_n}(x) ) ) { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| }}{ { \lfloor (\mu_\phi([w]) + 2\delta) |F_n|/ N \rfloor \choose \epsilon \cdot |F_n| }}. \end{align} $$

We now bound for any fixed $u \in S_n^{\prime }$ and each $u' \in A_u$ , $\sup _{x \in [u']} \phi _{F_n}(x)$ . First, note $u' = R_u^{v \rightarrow w}(S)$ for some $S \subset F_n$ . Since $E_X(v) \subset E_X(w)$ and by our choice of S, we know $E_X(u) \subset E_X(u')$ . We therefore know $\xi _{u, u'}$ (as defined above) is injective and non-identity on $[u]$ . In particular, we are replacing our v terms located at S with w terms just as we did in the construction of $u'$ . Note here, $\xi _{u, u'}([u]) \subset [u']$ and thus, we have

$$ \begin{align*}\sup_{x \in [u']} \phi_{F_n}(x) \geq \sup_{x \in \xi_{u, u'}([u])} \phi_{F_n}(x) = \sup_{x \in [u]} \phi_{F_n}(\xi_{u, u'}(x)). \end{align*} $$

We now fix any $x \in [u]$ . Note that since S is finite and left F-sparse, we may make the v to w replacements sequentially. Each of these replacements takes the form $\sigma _{g^{-1}} ( \xi _{v, w}( \sigma _g (x)) $ for some $g \in G$ . Note here, for each replacement of this kind, by definition of C, we have

$$ \begin{align*}\phi_{F_n}( \sigma_{g^{-1}} ( \xi_{v, w}( \sigma_g (x)) ) \geq \phi_{F_n}( x) - C. \end{align*} $$

Since we will make $|S|$ of these replacements, it follows that

$$ \begin{align*}\phi_{F_n}( \xi_{u, u'}(x)) \geq \phi_{F_n}(x) - |S| C. \end{align*} $$

We therefore know that

$$ \begin{align*}\sum_{u \in S_n^{\prime}} \sum_{u' \in A_u} \sup_{x \in [u']} \mathrm{exp }( \phi_{F_n}(x) ) \geq \sum_{u \in S_n^{\prime}} |A_u| \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) - |S| C). \end{align*} $$

Since $|A_u| \geq 1$ , we therefore have

(3) $$ \begin{align} \sum_{u \in S_n^{\prime}} \sum_{u' \in A_u} \sup_{x \in [u']} \mathrm{exp }( \phi_{F_n}(x) ) \geq \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) - |S| C). \end{align} $$

Combining equations (2) and (3), we arrive at our desired result that $\sum _{u \in L_n} \sup _{x \in [u]} \mathrm {exp }( \phi _{F_n}(x) )$ is bounded below by equation (1).

We will conclude the proof of Theorem 1.1 by computing a lower bound on topological pressure and arrive at a contradiction. Since $L_n \subset L_{F_n}(X)$ , it must be the case that

$$ \begin{align*} P_{\mathrm{top}}(\phi) \geq \limsup_{n \rightarrow \infty} |F_n|^{-1} \log \bigg( \sum_{u \in L_n} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) ) \bigg). \end{align*} $$

By application of Lemma 3.4, we can see that $P_{\mathrm {top}}(\phi )$ is bounded below by

$$ \begin{align*}\liminf_{n \rightarrow \infty} |F_n|^{-1} \log \frac{( \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) - ({ \epsilon \cdot |F_n| }/{ N}) C ) ) { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| }}{ { \lfloor (\mu_\phi([w]) + 2\delta) |F_n|/ N \rfloor \choose \epsilon \cdot |F_n| }}, \end{align*} $$

which is exactly equal to

(4) $$ \begin{align} &\liminf_{n \rightarrow \infty} |F_n|^{-1} \log \bigg( \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) ) \bigg) - \epsilon \cdot C\nonumber\\& \quad + |F_n|^{-1} {\kern-1pt}\log{\kern-1pt} { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| } - |F_n|^{-1} {\kern-1pt}\log{\kern-1pt} {\lfloor(\mu_\phi([w]) + 2\delta) |F_n|/ N \rfloor \choose\epsilon \cdot |F_n| }. \end{align} $$

We now note by our choice of a, b, and $\epsilon $ ,

$$ \begin{align*}\lim_{n \rightarrow \infty} |F_n|^{-1} \log { \lceil (\mu_\phi([v]) - 5\delta / 4) |F_n| /N \rceil \choose \epsilon \cdot |F_n| } -|F_n|^{-1} \log { \lfloor (\mu_\phi([w]) + 2\delta) |F_n|/ N \rfloor \choose \epsilon \cdot |F_n| }\end{align*} $$

$$ \begin{align*}\geq \lim_{n \rightarrow \infty} |F_n|^{-1} \log { a |F_n| \choose \epsilon \cdot |F_n| } -|F_n|^{-1} \log { b |F_n| \choose \epsilon \cdot |F_n| }> \epsilon C .\end{align*} $$

It therefore follows that equation (4) is strictly greater than

$$ \begin{align*}\liminf_{n \rightarrow \infty} |F_n|^{-1} \log \bigg( \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) ) \bigg) , \end{align*} $$

and since equation (4) is a lower bound for pressure, we therefore know

$$ \begin{align*}P_{\mathrm{top}}(\phi)> \liminf_{n \rightarrow \infty} |F_n|^{-1} \log \bigg( \sum_{u \in S_n^{\prime}} \sup_{x \in [u]} \mathrm{exp }( \phi_{F_n}(x) ) \bigg). \end{align*} $$

This contradicts Lemma 3.3. In particular, this means our assumption that $\mu _\phi ([v])> \mu _\phi ([w]) \cdot e^C$ was incorrect, arriving at our desired result that if $E_X(v) \subset E_X(w)$ , then

$$ \begin{align*}\mu_\phi([v]) \leq \mu_\phi([w]) \cdot \sup_{x \in [v]} \mathrm{exp }\bigg( \sum_{g \in G} \phi(\sigma_g(x)) - \phi(\sigma_g(\xi_{v, w}(x))) \bigg). \end{align*} $$

3.4 Locally constant corollaries

We will now explore the case where $\phi $ is locally constant.

The following result shows that when the finite configurations v and w agree on a sufficiently chosen boundary, we may compute bounds for their relative measure using only information from the configurations themselves. Further, when they have equal extender sets, we have a closed form formula depending only on $\phi $ , v, and w to compute the ratio of their measures.

We will say that $\phi $ is an H-potential for some $H \Subset G$ if for all $x, y \in X$ such that $x_H = y_H$ , then $\phi (x) = \phi (y)$ , that is, $\mathrm {Var}_H(\phi ) = 0$ . Additionally, for any $H \subset G$ , we will denote $H^\pm = H \cup H^{-1}$ .

Corollary 3.5. Let $H, F \Subset G$ , $v, w \in L_F(X)$ , and $\phi $ be an H-potential. Suppose that $E_X(v) \subset E_X(w)$ and for all $g \in F^c H^\pm \cap F$ , $v_g = w_g$ (that is, v and w agree on their H-boundary). Then, for any equilibrium state $\mu _\phi $ for $\phi $ ,

$$ \begin{align*} \mu_\phi([v]) \leq \mu_\phi([w]) \cdot \mathrm{exp }\bigg( \sum_{g \in F \backslash F^c H^{-1}} \phi ( \sigma_g (v)) - \phi ( \sigma_g(\xi_{v, w}(w))) \bigg). \end{align*} $$

Proof. We first note, for all $x \in [v]$ and for any $g \in F \cap F^c H^{-1}$ , by assumption, we have $x_{gH} = \xi _{v, w}(x)_{gH}$ . Since $\phi $ is an H-potential, it immediately follows that for all $g \in F \cap F^c H^{-1}$ ,

$$ \begin{align*} \phi ( \sigma_g (x)) - \phi ( \sigma_g(\xi_{v, w}(x))) = 0. \end{align*} $$

We now consider $g \in F^c$ . If $gH \cap F = \emptyset $ , then $x_{gH} = \xi _{v, w}(x)_{gH}$ by definition of $\xi _{v, w}$ . We now consider $gH \cap F \neq \emptyset $ . However, by assumption, since $g \in F^c$ , we know for all $f \in gH \cap F$ , $v_f = w_f$ and again we can conclude $x_{gH} = \xi _{v, w}(x)_{gH}$ . Thus, we know for all $g \in F^c$ ,

$$ \begin{align*} \phi \circ \sigma_g (x) - \phi \circ \sigma_g(\xi_{v, w}(x)) = 0. \end{align*} $$

Since $\phi $ is locally constant, it has summable variation, and we can apply Theorem 1.1 to see

$$ \begin{align*}\mu_\phi([v]) \leq \mu_\phi([w]) \sup_{x \in [v]} \mathrm{exp }\bigg( \sum_{g \in F \backslash F^c H^{-1}} \phi ( \sigma_g (x)) - \phi ( \sigma_g(\xi_{v, w}(x))) \bigg). \end{align*} $$

We let $x, y \in [v]$ , $g\in F \backslash F^c H^\pm $ , and $h \in H$ . By construction, we know $g \notin F^c H^{-1}$ and, thus, $gH \cap F^c = \emptyset $ . Since $\phi $ is an H-potential, it follows that $\phi \circ \sigma _g (x) = \phi \circ \sigma _g (y)$ , and similarly for $\xi _{v, w}(x)$ and $\xi _{v, w}(y)$ . We can therefore conclude that $\sum _{g \in F \backslash F^c H^\pm } \phi ( \sigma _g (x)) - \phi ( \sigma _g(\xi _{v, w}(x)))$ does not depend on the choice of $x \in [v]$ and can be computed by only looking at the cylinder set, and we have arrived at our desired result.

As an immediate corollary, we have the following.

Corollary 3.6. Let $H, F \Subset G$ , $v, w \in L_F(X)$ , and $\phi $ be an H potential. Suppose that $E_X(v) = E_X(w)$ and for all $g \in F^c H^\pm \cap F$ , $v_g = w_g$ (that is, v and w agree on their H-boundary). Then, for any equilibrium state $\mu _\phi $ for $\phi $ ,

$$ \begin{align*}\frac{\mu_\phi([v])}{\mathrm{exp } \big( \sum_{g \in F \backslash F^c H^{-1}} \phi \circ \sigma_g (v) \big)} = \frac{\mu_\phi([w])}{\mathrm{ exp } \big( \sum_{g \in F \backslash F^c H^{-1}} \phi \circ \sigma_g (w) \big)}. \end{align*} $$

We remind the reader that in the case of $\phi = 0$ , equilibrium states correspond with measures of maximal entropy. In this case, as a corollary of Theorem 1.1, we extend the results of García-Ramos and Pavlov in [Reference García-Ramos and Pavlov11] to the countable amenable subshift setting.

4.1 Homoclinic relation and conformal Gibbs

As previously mentioned, a measure $\mu $ is Gibbs for a potential $\phi $ (in the sense of the theorem of Dobrušin and that of Lanford–Ruelle) if and only if it is conformal Gibbs for $\phi $ . This section will explore our conformal Gibbs-like result, for which we will follow the definition found in [Reference Borsato and MacDonald3].

First, we define the homoclinic relation, also known as the Gibbs relation, as $\mathfrak {T}_X \subset X \times X$ such that $(x, y) \in \mathfrak {T}_X$ if and only if $x_{F^c} = y_{F^c}$ for some $F \Subset G$ . We now say a Borel isomorphism $f: X \rightarrow X$ is a holonomy of $\mathfrak {T}_X$ if for all $x \in X$ , $(x, f(x)) \in \mathfrak {T}_X$ . It is known (see [Reference Borsato and MacDonald3]) that there exists a countable group $\Gamma $ of holonomies of X such that $\mathfrak {T}_X = \{ (x, \gamma (x)) : x \in X, \gamma \in \Gamma \}$ . In fact, by a trivial extension of [Reference Borsato and MacDonald3, Lemma 1], we can see immediately that $\Gamma = \langle \xi _{v, w} : v, w \in L_F(X), F \Subset G \rangle$ is a countable group of holonomies such that $\mathfrak {T}_X = \{ (x, \gamma (x)) : x \in X, \gamma \in \Gamma \}$ .

For any measurable $A \subset X$ , we define $\mathfrak {T}_X(A) = \bigcup _{x \in A} \{ y \in X : (x, y) \in \mathfrak {T}_X \}$ . For a Borel measure $\mu $ on X, we say $\mu $ is $\mathfrak {T}_X$ -non-singular if for all null sets $A \subset X$ , $\mu (\mathfrak {T}_X(A)) = 0$ . Note for any holonomy $f:X \rightarrow X$ , if $A \subset X$ is a null set and $\mu $ is $\mathfrak {T}_X$ -non-singular, then $\mu \circ f(A) \leq \mu (\mathfrak {T}_X(A)) = 0$ and therefore $\mu \circ f << \mu $ .

We define a cocycle on $\mathfrak {T}_X$ to be any measurable $\psi : \mathfrak {T}_X \rightarrow \mathbb {R}$ such that for all $x, y, z \in X$ with $(x,y), (y, z) \in \mathfrak {T}_X$ , we have $\psi (x, y) + \psi (y, z) = \psi (x, z)$ . We can now define conformality following [Reference Borsato and MacDonald3, Definition 1].

Definition 4.1. Let $\mu $ be a $\mathfrak {T}_X$ -non-singular Borel probability measure on X and let $\psi : \mathfrak {T}_X \rightarrow \mathbb {R}$ be a cocycle. We say $\mu $ is $(\psi , \mathfrak {T}_X)$ -conformal if for any holonomy $f: X \rightarrow X$ , we have

$$ \begin{align*}\frac{ d\mu \circ f}{d\mu}(x) = \mathrm{exp }( \psi(x, f(x) ). \end{align*} $$

Since we are concerned with Gibbs measures, in general, for a potential $\phi \in SV(X)$ , we define the following cocycle:

$$ \begin{align*}\psi_\phi (x, y) = \sum_{g \in G} \phi \circ \sigma_g (y) - \phi \circ \sigma_g(x)\end{align*} $$

and we will say that a measure $\mu $ is conformal Gibbs for $\phi $ if it is $(\psi _\phi , \mathfrak {T}_X)$ -conformal.

Connecting this concept further to the classical notion of Gibbs measures, it was shown in [Reference Borsato and MacDonald3] that a measure is conformal Gibbs for $\phi $ if and only if it is Gibbs for $\phi $ in the sense of Definition 2.8.

In addition to the homoclinic relation, in [Reference Meyerovitch17], Meyerovitch defined a subrelation as follows. First, let

$$ \begin{align*}\mathcal{F}(X)= \{ f \in \mathrm{Homeo}(X) : \text{ there exists } F \Subset G \; \text{ such that for all } x \in X, \; f(x)_{F^c} = x_{F^c} \}.\end{align*} $$

In particular, $\mathcal {F}(X)$ is the group of all background preserving homeomorphisms. We then define $\mathfrak {T}_X^0 = \{ (x, f(x)) : x \in X, f \in \mathcal {F}(X) \}$ . Clearly, $\mathfrak {T}_X^0 \subset \mathfrak {T}_X$ ; however, in general, these are not necessarily equal.

Notice that when X is an SFT, then each $\xi _{ v, w}$ is in fact a homeomorphism and thus $\xi _{v, w} \in \mathcal {F}(X)$ . In this case, $\mathfrak {T}_X^0 = \mathfrak {T}_X$ . Additionally, even if X is not an SFT, if $E_X(v) = E_X(w)$ , $\xi _{v, w}$ is again a homeomorphism. However, it need not be the case that $\xi _{ v, w}$ is continuous in general.

Example 4.2. Take $X \subset \{ 0, 1\}^{\mathbb {Z}}$ to the orbit closure of $x = 0^\infty 1 0^\infty $ (which is sometimes referred to as the sunny-side-up subshift). We define $\xi _{1, 0}$ to swap a $1$ and $0$ in the identity location, and note here that $E_X(1) \subsetneq E_X(0)$ . For each $n \geq 0$ , define $x_n \in X$ such that $(x_n)_n = 1$ . Then, for all $n \geq 1$ , $\xi _{1, 0}(x_n) = x_n$ . However, we have $x_n \rightarrow 0^\infty $ and $\xi _{1, 0}(0^\infty ) = x_0$ . It immediately follows that

$$ \begin{align*}\lim_{n \rightarrow \infty} \xi_{1, 0}(x_n) = 0^\infty \neq x_0 = \xi_{1, 0}( \lim_{n \rightarrow \infty} x_n) \end{align*} $$

and we can conclude $\xi _{1, 0}$ is not continuous.

In the case of $G = \mathbb {Z}^d$ and $\phi \in C(X)$ has d-summable variation, Meyerovitch proved the following result.

In particular, $\mu _\phi $ is conformal Gibbs with respect to the sub-relation $\mathfrak {T}_X^0$ . What this means in the terminology of our paper is for $G = \mathbb {Z}^d$ and $\phi $ with d-summable variation, if $E_X(v) = E_X(w)$ , then for $\mu _\phi $ -a.e. $x \in X$ , $({d (\mu \circ \xi _{v, w} ) }/{d\mu })(x) = \mathrm {exp }( \psi _\phi (x, \xi _{v, w}(x)) )$ . We will use Theorem 1.1 to extend this result both to the general amenable group setting as well as to provide an inequality on the derivative when $E_X(v) \subset E_X(w)$ .

4.2 Proof of conformal Gibbs result

In this section, we will prove the following theorem.

Theorem 4.4. Let $F \Subset G$ , $v, w \in L_F(X)$ , $\phi \in SV(X)$ , and $\mu _\phi $ be an equilibrium state for $\phi $ . If $E_X(v) \subset E_X(w)$ , then $\mu _\phi \circ \xi _{v, w}$ is absolutely continuous with respect to $\mu _\phi $ when restricted to $[w]$ and for $\mu _\phi $ -a.e. $x \in [w]$ ,

$$ \begin{align*}\frac{ d (\mu \circ \xi_{v, w} ) }{d \mu}(x) \leq \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ). \end{align*} $$

For the remainder of this section, we fix some $F \Subset G$ , $v, w \in L_F(X)$ , $\phi \in SV(X)$ , and $\mu _\phi $ an equilibrium state for $\phi $ .

Proof. Let $A \subset [w]$ be a null set. Note here

$$ \begin{align*}\xi_{v, w}(A) = \{ x \in A : \xi_{v, w}(x) = x\} \sqcup \{ \xi_{v, w}(x) : x \in A \text{ and } \xi_{v, w}(x) \in [v] \}. \end{align*} $$

Since $ \{ x \in A : \xi _{v, w}(x) = x\} \subset A$ , it must be a null set, and thus it is sufficient to assume without loss of generality that $\xi _{v, w}(A) \cap A = \emptyset $ .

Since we know $\mu _\phi $ is outer regular, we can let $U_n =\bigcup _{ i \leq N_n} [w_{n, i}]$ be an outer approximation by cylinder sets such that $\mu _\phi (A) = \lim _{n \rightarrow \infty } \mu _\phi (U_n)$ . By taking sufficiently large n, we can assume without loss of generality that $[w_{n, i}] \subset [w]$ for all $n, i$ . Note for each $n, i$ , we have $E_X(\xi _{v, w}(w_{n, i})) \subset E_X(w_{n, i})$ . We now apply Theorem 1.1 with respect to $\phi $ , $\mu _\phi $ , $v_{n, i}$ , and $w_{n, i}$ to see

$$ \begin{align*}\mu_\phi(\xi_{v, w}(A)) &= \lim_{n \rightarrow \infty} \sum_{i \leq N_n} \mu_\phi ( \xi_{v, w}([w_{n, i}]) )\\ &\leq \lim_{n \rightarrow \infty} \sum_{i \leq N_n} \mu_\phi ([w_{n, i}]) \sup_{x \in [v]} \mathrm{exp } (f(x)) = 0.\\[-3.9pc] \end{align*} $$

In particular, this means we can discuss the LRN derivative when restricted to $[w]$ . We now define for each $n \in \mathbb {N}$ , $\phi _n : X \rightarrow \mathbb {R}$ to be an $E_n$ potential approximating $\phi $ from above. In particular, for all $u \in L_{E_n}(X)$ , for any $x \in [u]$ , let $\phi _n(x) = \sup _{y \in [u]} \phi (y)$ .

Proof. First note, for any $k, n \in \mathbb {N}$ , we have

$$ \begin{align*} \mathrm{Var}_{E_k}(\phi_n - \phi) \leq 2 || \phi||_\infty \quad\text{and}\quad \mathrm{Var}_{E_k}(\phi_n - \phi) \leq 2 \mathrm{Var}_{E_k}(\phi) .\end{align*} $$

Let $\epsilon> 0$ and note since $\phi $ has summable variation according to $\{ E_n \}$ , by definition, there exists some $M \in \mathbb {N}$ such that $\sum _{k=M+1}^\infty |E_{k+1} \backslash E_k| \mathrm {Var}_{E_k}( \phi ) < \epsilon $ . We fix this M. We now note by continuity of $\phi $ and since $\{ E_n\}$ is an exhaustive sequence, we know there exists some $N \in \mathbb {N}$ such that for all $n \geq N$ ,

$$ \begin{align*} || \phi_N - \phi||_\infty < \epsilon \bigg( 2 |E_1| + \sum_{k=1}^M |E_{k+1} \backslash E_k| \bigg)^{-1}. \end{align*} $$

We now let $n \geq N$ and we examine

$$ \begin{align*}|| \phi_N - \phi||_{SV_{\{ E_n \}}} &= 2 |E_1| \cdot || \phi_n - \phi||_\infty + \sum_{k=1}^M |E_{k+1} \backslash E_k| \mathrm{Var}_{E_k}(\phi_n - \phi)\\&\quad + \sum_{k=M+1}^\infty |E_{k+1} \backslash E_k| \mathrm{Var}_{E_k}(\phi_n - \phi)\\&\leq || \phi_n - \phi||_\infty \bigg(\! 2 |E_1| + \sum_{k=1}^M |E_{k+1} \backslash E_k| \!\bigg) + 2 \sum_{k=M+1}^\infty\! |E_{k+1} \backslash E_k| \mathrm{ Var}_{E_k}( \phi).\end{align*} $$

By our choice of N and M, we know for all $n \geq N$ , $||\phi _N - \phi ||_{SV_{\{ E_n \}}} < 3 \epsilon $ . Since $\epsilon $ was arbitrary, we can conclude the proof of the observation.

Proof. Using our locally constant approximations $\phi _n$ , we define for each $n \in \mathbb {N}$ ,

$$ \begin{align*}f_n(x) = \psi_{\phi_n}(x, \xi_{v, w}(x) ) = \sum_{g \in G} \phi_n(\sigma_g(\xi_{v, w}(x))) - \phi_n(\sigma_g(x)). \end{align*} $$

Since $\phi _n$ is locally constant, we know the infinite sum is in fact a finite sum and so $f_n$ is continuous on $[v]$ . We now claim that $f_n \rightarrow f$ uniformly on $[v]$ . By Observation 3.12, we know for all $x \in [v]$ ,

$$ \begin{align*}|f_n(x) - f(x)| \leq |F| \cdot ||\phi_n - \phi||_{SV_{\{ E_n \}}}. \end{align*} $$

Since $\phi _n \rightarrow \phi $ with respect to the summable variation norm by Observation 4.6, we can conclude that $f_n \rightarrow f$ uniformly on $[v]$ . It immediately follows that f is continuous on $[v]$ .

We can combine the above observations with Theorem 1.1 to prove the desired result.

Theorem 4.3. Let $F \Subset G$ , $v, w \in L_F(X)$ , $\phi \in SV(X)$ , and $\mu _\phi $ be an equilibrium state for $\phi $ . If $E_X(v) \subset E_X(w)$ , then for $\mu _\phi $ -a.e. $x \in [w]$ ,

$$ \begin{align*}\frac{ d\mu \circ \xi_{v, w}}{d \mu}(x) \leq \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ). \end{align*} $$

Proof. First, we fix $F \Subset G$ , $v, w \in L_F(X)$ , $\phi \in C(X)$ , and $\mu _\phi $ as in the theorem. We again define for all $x \in X$ ,

$$ \begin{align*}f(x) = \psi_\phi(x, \xi_{v, w}(x)) = \sum_{g \in G} \phi(\sigma_g(\xi_{v,w}(x))) - \phi(\sigma_g(x)),\end{align*} $$

which we know to be continuous on $[v]$ by Observation 4.7.

We fix any Følner sequence $\{ F_n \}$ and assume without loss of generality that $F \subset F_n$ for all n. Fix any $x \in [w]$ . We define $w_n = x_{F_n}$ and $v_n = \xi _{v, w}(x)_{F_n}$ . Note here $\bigcap _{n\in \mathbb {N}} [w_n] = x$ , $\bigcap _{n \in \mathbb {N}} [v_n] = \xi _{v, w}(x)$ . Since $F \subset F_n$ , we know $E_X(v_n) \subset E_X(w_n)$ . We can directly apply Theorem 1.1 to see for each $n \in \mathbb {N}$ ,

$$ \begin{align*}\mu_\phi(v_n) \leq \mu_\phi(w_n) \sup_{y \in [v_n]} \mathrm{exp }( - f(y) ). \end{align*} $$

Since by Observation 4.7, we know f is continuous on $[v]$ and thus there exists some $y_n \in [v_n]$ such that $\sup _{y \in [v_n]} \mathrm {exp }( - f(y) ) = \mathrm {exp }( - f(y_n) )$ . We therefore have

$$ \begin{align*}\frac{\mu_\phi(v_n)}{\mu_\phi(w_n)} \leq \mathrm{exp }( - f(y_n) ). \end{align*} $$

Since $y_n \in [v_n]$ and $\bigcap _{n \in \mathbb {N}} [v_n] = \{ \xi _{v, w}(x) \}$ , we know $y_n \rightarrow \xi _{v, w}(x)$ . We can now take limits to see

$$ \begin{align*}\frac{d \mu_\phi \circ \xi_{v, w}}{d \mu_\phi} (x) = \lim_{n \rightarrow \infty} \frac{\mu_\phi(v_n)}{\mu_\phi(w_n)} \leq \lim_{n \rightarrow \infty} \mathrm{exp }( - f(y_n) ) = \mathrm{exp }( f(x)).\\[-41pt] \end{align*} $$

Finally, we note that whenever $\mu _\phi $ satisfies the equations in Theorem 1.3, it is trivial to show $\mu _\phi $ must also satisfy the equations in Theorem 1.1.

Observation 4.4. Let X be a subshift and $u, v \in L_F(X)$ such that $E_X(v) \subset E_X(w)$ . Let $\phi \in C(X)$ and $\mu _\phi $ satisfy for $\mu _\phi $ -a.e. $x \in [w]$ ,

$$ \begin{align*}\frac{ d (\mu \circ \xi_{v, w} ) }{d \mu}(x) \leq \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ). \end{align*} $$

Then,

$$ \begin{align*}\mu_\phi([v]) \leq \mu_\phi([w]) \cdot \sup_{x \in [v]} \mathrm{exp }\bigg( \sum_{g \in G} \phi ( \sigma_g(x)) - \phi(\sigma_g(\xi_{v, w}(x))) \bigg). \end{align*} $$

Proof. First, we notice that since $E_X(v) \subset E_X(w)$ , then $[v] \subseteq \xi _{v,w}([w])$ . Further, since $\xi _{v, w}$ is an involution, we know

$$ \begin{align*}\mu_\phi([v]) = \mu_\phi( \xi_{v, w}( \xi_{v, w}([v]))) = \int_{x \in \xi_{v, w}([v])} \frac{d (\mu_\phi \circ \xi_{v, w} ) }{d \mu_\phi }(x)\, d\mu_\phi .\end{align*} $$

Since by assumption, for $\mu _\phi $ -a.e. $x \in [w]$ ,

$$ \begin{align*}\frac{ d (\mu \circ \xi_{v, w} ) }{d \mu}(x) \leq \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ), \end{align*} $$

we know that

$$ \begin{align*}\int_{x \in \xi_{v, w}([v])} \frac{d (\mu_\phi \circ \xi_{v, w} ) }{d \mu_\phi }(x) \,d\mu_\phi \leq \int_{x \in \xi_{v, w}([v])} \mathrm{exp }( \psi_\phi( \xi_{v,w}(x), \xi_{v,w}(\xi_{v, w}(x))) )\, d\mu_\phi. \end{align*} $$

It follows immediately that

$$ \begin{align*}\mu_\phi([v]) \leq \mu_\phi( \xi_{v, w}([v])) \cdot \sup_{y \in [v]} \mathrm{exp }( \psi_\phi( \xi_{v,w}(x), x) ). \end{align*} $$

Since $\xi _{v, w}([v]) \subset [w]$ , we arrive at our desired conclusion.

We will now note that as a corollary of Theorem 1.3, we can extend [Reference Meyerovitch17, Theorem 3.1 and Corollary 3.2] to the countable amenable group subshift setting. We have become aware that [Reference Barbieri and Meyerovitch2, Theorem B] extends this further to the countable sofic group subshift setting.

Proof. First note, as observed in [Reference Meyerovitch17], the collection $\{ \xi _{v, w} : E_X(v) = E_X(w) \}$ generates $\mathcal {F}(X)$ . By application of the results in [Reference Meyerovitch17], it is therefore sufficient to show that for any $v, w \in L_F(X)$ with $E_X(v) = E_X(w)$ , we have for all $x \in X$ ,

$$ \begin{align*}\frac{ d\mu \circ \xi_{v, w}}{d \mu}(x) = \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ). \end{align*} $$

Let $v, w \in L_F(X)$ such that $E_X(v) = E_X(w)$ . Let $x \in [w]$ and for all $n \in \mathbb {N}$ , define $v_n, w_n \in L_{F_n}(X)$ as in the proof of Theorem 1.3. By application of Theorem 1.1, we can see that

$$ \begin{align*}\inf_{y \in [v_n]} \mathrm{exp }( - f(y) ) \leq \frac{\mu_\phi([v_n])}{\mu_\phi([w_n])} \leq \sup_{y \in [v_n]} \mathrm{exp }( - f(y) ). \end{align*} $$

By taking limits, we can conclude that for all $x \in [w]$ ,

$$ \begin{align*}\frac{ d\mu \circ \xi_{v, w}}{d \mu}(x) = \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ). \end{align*} $$

The proof is similar for $x \in [v]$ and for all $x \in X \backslash ([v] \cup [w])$ , we know

$$ \begin{align*}\frac{ d\mu \circ \xi_{v, w}}{d \mu}(x) = \mathrm{exp }( \psi_\phi( x, \xi_{v, w}(x)) ) = 1.\end{align*} $$

Again, by following the techniques in [Reference Meyerovitch17], it follows that $\mu _\phi $ is $(\mathfrak {T}_X^0, \psi _\phi )$ -conformal.