2 Countable state Markov shifts

Let S be a countable set and let A be a matrix indexed by $S \times S$ with entries in $\{0,1\}$ . We call S the set of states and A the transition matrix. We write $A(s,s')$ for the $(s,s')$ entry of A. We then let

$$ \begin{align*} \Sigma=\{x=(x_n)_{n=0}^\infty \text{ : } x_n \in S \text{ and } A(x_n,x_{n+1})=1 \ \forall n \in \mathbb Z^+\}. \end{align*} $$

Let $\sigma : \Sigma \to \Sigma $ be the shift map defined by $(\sigma x)_n=x_{n+1}$ . Then, we call $(\Sigma ,\sigma )$ a (one-sided) countable state Markov shift. If S is finite, then $(\Sigma ,\sigma )$ is called a (one-sided) subshift of finite type.

A word $w=(w_1,\ldots ,w_{n}) \in S^n$ is called allowed if $A(w_i,w_{i+1})=1$ for all ${i=~1,\ldots ,n-1}$ and we call $[w]=\{x \in \Sigma \text { : } x_i=w_i \ \forall i=1,\ldots ,n\}$ the cylinder defined by w. Let $\mathcal W_n \subset S^n$ denote the set of all allowed words of length n and let $\mathcal W := \bigcup _{n=1}^\infty \mathcal W_n$ . For $a \in S$ , we write

$$ \begin{align*} \mathcal W_n(a) := \{w=(w_1,\ldots,w_n)\in \mathcal W_n \text{ : } w_1=a \text{ and } A(w_n,a)=1\}. \end{align*} $$

If w is a finite word and x is either a finite word or element of $\Sigma $ , then we let $wx$ denote the concatenation.

The topology on $\Sigma $ is the topology generated by cylinder sets. The shift map is continuous and we have that $\Sigma $ is compact if and only if S is finite. We say that $\sigma : \Sigma \to \Sigma $ is topologically transitive if there is a point with a dense orbit. We say that $\sigma : \Sigma \to \Sigma $ is topologically mixing if for all $a,b \in S$ , there exists $N(a,b)$ such that for all $n \ge N(a,b)$ , there exists $w \in \mathcal W_n$ with $awb \in \mathcal W$ . (This is equivalent to the usual definition of topological mixing in topological dynamical systems.)

We say that $\varphi : \Sigma \to \mathbb R$ is locally Hölder continuous if

$$ \begin{align*} \sup_{w \in \mathcal W_n} \sup_{x,y \in [w]} |\varphi(x)-\varphi(y)| \le C\theta^n \end{align*} $$

for some $C>0$ and $0<\theta <1$ , for all $n \ge 1$ . If we wish to specify $\theta $ , then we say that $\varphi $ is $\theta $ -locally Hölder continuous. We write

$$ \begin{align*} \varphi^n(x) := \varphi(x) +\varphi(\sigma x) + \cdots + \varphi(\sigma^{n-1}x). \end{align*} $$

A locally Hölder function satisfies the following estimates.

Lemma 2.1. [Reference Sarig29, Lemma 1]

Let $\varphi : \Sigma \to \mathbb R$ be locally Hölder continuous. Then, for all $a \in S$ and all $o,o' \in [a]$ , there exists $B>0$ such that

$$ \begin{align*} |\varphi^n(wo)-\varphi^n(wo')| \le B \end{align*} $$

for all $w \in \mathcal W_n$ such that $wa \in \mathcal W_{n+1}$ .

Assume that $\sigma $ is topologically transitive and that $\varphi : \Sigma \to \mathbb R$ is locally Hölder continuous. Then, we can define the Gurevič pressure $P_{\mathrm {G}}(\varphi ,\sigma ) \in \mathbb R \cup \{\infty \}$ by

$$ \begin{align*} P_{\mathrm{G}}(\varphi,\sigma) = \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\sigma^nx=x} e^{\varphi^n(x)} \mathbb 1_{[a]}(x) \end{align*} $$

for any choice of $a \in S$ [Reference Sarig29]. If $\sigma $ is topologically mixing, then the limsup may be replaced by a limit.

The transfer operator $L_\varphi $ acts on a real-valued function on $\Sigma $ by the formula

$$ \begin{align*} (L_\varphi v)(x) = \sum_{\sigma y=x} e^{\varphi(y)} v(y) \end{align*} $$

(which may be infinite). We say that $\varphi $ is summable if $\|L_\varphi 1\|_\infty <\infty $ . For later use, we note that, using the local Hölder continuity of $\varphi $ , this is equivalent to the condition

$$ \begin{align*} \sum_{a \in S} \exp(\sup\{\varphi(y) \text{ : } y \in [a]\})<\infty \end{align*} $$

used in [Reference Kesseböhmer and Kombrink18, Reference Mauldin and Urbański21, Reference Mauldin and Urbański22]. If $\varphi $ is summable, then $P_{\mathrm {G}}(\varphi ,\sigma )<\infty $ .

We may also characterize the Gurevič pressure directly in terms of the transfer operator.

Lemma 2.2. [Reference Sarig31, Theorem 4.4]

If $P_{\mathrm {G}}(\varphi ,\sigma )<\infty $ , and $v : \Sigma \to \mathbb R$ is continuous, non-negative, not identically zero and with support contained inside a finite union of cylinders, then

$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{n} \log (L_\varphi^n v)(x) =P_{\mathrm{G}}(\varphi,\sigma) \end{align*} $$

for all $x \in \Sigma $ . If the shift is topologically mixing, then the limsup may be replaced by a limit.

3 Skew product extension

We now consider skew product extensions. Let G be a finitely generated countable group (with the discrete topology), with identity e. Suppose we are given a function $\psi : \Sigma \to G$ with $\psi ((x_i)_{i=1}^\infty ) = \psi (x_1)$ . (Any continuous function from $\Sigma $ to G is locally constant and can be reduced to this form by recoding.) Then, we can define the skew product extension

$$ \begin{align*} T_\psi : \Sigma \times G \to \Sigma \times G \end{align*} $$

by

$$ \begin{align*} T_\psi(x,g) = (\sigma x,g\psi(x)). \end{align*} $$

Note that

$$ \begin{align*} T_\psi^n(x,g) = (\sigma^nx,g\psi_n(x)), \end{align*} $$

where

$$ \begin{align*} \psi_n(x) = \psi(x) \psi(\sigma x) \cdots \psi(\sigma^{n-1}x). \end{align*} $$

We will always assume that $T_\psi $ is topologically transitive.

The system $T_\psi : \Sigma \times G \to \Sigma \times G$ is also a countable state Markov shift with set of states $S \times G$ and a transition matrix $\mathbb A$ defined by

$$ \begin{align*} \mathbb A((i,g),(j,h))=1 \quad \iff \quad A(i,j)=1 \text{ and } g\psi(i)=h. \end{align*} $$

Thus, Gurevič pressure is also defined for the extension.

Given an extension with group G, we will also need to consider the abelianized system as follows. Let $G^{\mathrm {ab}}$ denote the abelianization of G and let $\overline G$ denote the torsion-free part of $G^{\mathrm {ab}}$ , that is, $\overline {G}$ is the quotient of $G^{\mathrm {ab}}$ by its torsion subgroup. Then, $\overline {G}$ is isomorphic to $\mathbb Z^d$ for some $d \ge 0$ . We use additive notation and let $0$ to denote the identity. Let ${\pi :G \to \overline {G}}$ be the projection homomorphism. Define $\bar \psi : \Sigma \to \overline {G}$ by $\bar \psi = \pi \circ \psi $ ; then, we have the skew product $T_{\bar \psi } : \Sigma \times \overline {G} \to \Sigma \times \overline {G}$ . (If $d=0$ , then we can regard $T_{\bar \psi }$ as $\sigma $ .)

Given $\varphi : \Sigma \to \mathbb R$ , define $\tilde \varphi : \Sigma \times G \to \mathbb R$ by $\tilde \varphi (x,g) = \varphi (x)$ and $\bar \varphi : \Sigma \times \overline {G} \to \mathbb R$ by $\bar \varphi (x,k) = \varphi (x)$ . Following the convention in §1, we write all of these functions as $\varphi $ . The next result proves the inequalities (1.2) in §1.

Lemma 3.1. For all $w \in \mathbb R^d$ , we have

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_\psi) \le P_{\mathrm{G}}(\varphi,T_{\bar \psi}) \le P_{\mathrm{G}}(\varphi + \langle w,\bar \psi \rangle,\sigma). \end{align*} $$

Proof. Let $a \in S$ , then $[a] \times \{e\}$ is a cylinder set for $\Sigma \times G$ and $[a] \times \{0\}$ is a cylinder set for $\Sigma \times \overline {G}$ . It is clear that $T_\psi ^n(x,e) = (x,e)$ implies $T_{\bar \psi }^n(x,0)=(x,0)$ and, therefore,

$$ \begin{align*} \sum_{T_\psi^n(x,e)=(x,e)} e^{\varphi^n(x,e)} \mathbb 1_{[a] \times \{e\}}(x,e) \le \sum_{T_\psi^n(x,0)=(x,0)} e^{\varphi^n(x,0)} \mathbb 1_{[a] \times \{0\}}(x,0). \end{align*} $$

Hence, $P_{\mathrm {G}}(\varphi ,T_\psi ) \le P_{\mathrm {G}}(\varphi ,T_{\bar \psi })$ . By a similar argument, $P_{\mathrm {G}}(\varphi ,T_{\bar \psi }) \le P_{\mathrm {G}}(\varphi ,\sigma )$ . Furthermore, if $T_{\bar \psi }^n(x,0)=(x,0)$ , then $\sigma ^nx=x$ and $\bar \psi ^n(x)=0$ , so we see that

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_{\bar \psi}) = P_{\mathrm{G}}(\varphi + \langle w,\bar \psi \rangle,T_{\bar \psi}) \end{align*} $$

for all $w \in \mathbb R^d$ .

Each of the skew products has its own transfer operator (associated to $\varphi $ ), which we distinguish by using different fonts, as follows:

$$ \begin{align*} (\mathcal L_{\varphi}v)(x,g) := \sum_{T_\psi(y,h) = (x,g)} e^{\varphi(y)} v(y,h) = \sum_{\sigma y=x} e^{\varphi(y)} v(y,g\psi(y)^{-1}) \end{align*} $$

and

$$ \begin{align*} (\mathfrak L_{\varphi}v)(x,m) := \sum_{T_{\bar \psi}(y,k) = (x,m)} e^{\varphi(y)} v(y,k) = \sum_{\sigma y=x} e^{\varphi(y)} v(y,m-\bar \psi(y)). \end{align*} $$

The formulae for iterates are

$$ \begin{align*} (\mathcal L_{\varphi}^nv)(x,g) = \sum_{T_\psi^n(y,h)=(x,g)} e^{\varphi^n(y)} v(y,h) =\sum_{\sigma^n y=x} e^{\varphi^n(y)} v(y,g\psi_n(y)^{-1}) \end{align*} $$

and

$$ \begin{align*} (\mathfrak L_{\varphi}^n v)(x,m) = \sum_{T_{\bar \psi}^n(z,k) = (x,m)} e^{\varphi(y)} v(z,k) = \sum_{\sigma y=x} e^{\varphi(y)} v(y,m-\bar \psi_n(y)). \end{align*} $$

4 Amenability implies equality

In the section, we prove Theorem 1.3. Since we know from (1.2) that $P_{\mathrm {G}}(\varphi ,T_\psi ) \le P_{\mathrm {G}}(\varphi ,T_{\bar \psi })$ , we only need to prove that $P_{\mathrm {G}}(\varphi ,T_{\bar \psi }) \le P_{\mathrm {G}}(\varphi ,T_\psi )$ . To simplify the notation, we shall write $\rho = e^{P_{\mathrm {G}}(\varphi ,T_\psi )}$ .

We begin by outlining the strategy employed in the proof. We shall work with the operator $\mathcal L_\varphi $ , but we only need to consider functions defined on a countable set $\mathcal Z \times G$ , where $\mathcal Z$ is the set of all pre-images (under $\sigma $ ) of a prescribed ‘base point’. The first task is to construct a bounded positive function $\Phi : \mathcal Z \times G \to \mathbb R^{>0}$ from limits of ratios of what one might think of as ‘Green’s functions’ $\zeta _{z,g}(t)$ , with a parameter decreasing to $\rho $ . This is carried out in Lemmas 4.2 and 4.3. Then, in Lemma 4.4, we show that $\mathcal L_\varphi \Phi = \rho \Phi $ . The next step, in Lemma 4.6, is to replace $\Phi $ with a function ${\phi : \mathcal Z \times G \to \mathbb R^{>0}}$ such that $\mathcal L_\varphi ^n\phi \le \rho ^n \phi $ and $\phi (z,\cdot )/\phi (z,e)$ is always a homomorphism from G to $\mathbb R^{>0}$ . This is the only point in the proof where the amenability of G is used: we consider an invariant mean on $\ell ^\infty (G,\mathbb R)$ applied to a bounded sequence derived from $\Phi $ , along with Jensen’s inequality. The proof of Theorem 1.3 is then nearly complete. In Lemma 4.7, using the homomorphism property, $\phi $ descends to a function $\bar \phi : \mathcal Z \times \overline {G} \to \mathbb R^{>0}$ satisfying ${\mathfrak L_\varphi ^n\bar \phi \le \rho ^n\bar \phi }$ . Finally, Lemma 2.2 relates $P_{\mathrm {G}}(\varphi ,T_{\bar \psi })$ to the growth rate of the iterates of $\mathfrak L_\varphi $ and, hence, we obtain $P_{\mathrm {G}}(\varphi ,T_{\bar \psi }) \le \log \rho $ .

Fix $a \in S$ ; for $o \in [a]$ and $g \in G$ , we will work with the quantity

$$ \begin{align*} \sum_{\substack{\sigma^n x=o \\x_1=a\\ \psi_n(x)=g}} e^{\varphi^n(x)} = \sum_{\substack{\sigma^n y=o}} e^{\varphi^n(y)} \mathbb 1_{[a] \times \{e\}}(x,g\psi_n(x)^{-1}) = (\mathcal L_{\varphi}^n \mathbb 1_{[a] \times \{e\}})(o,g). \end{align*} $$

Since $T_\psi $ is topologically transitive, it follows from Lemma 2.2 that

(4.1) $$ \begin{align} \limsup_{n \to \infty} \frac{1}{n} \log (\mathcal L_{\varphi}^n \mathbb 1_{[a] \times \{e\}})(o,g) = P_{\mathrm{G}}(\varphi,T_\psi). \end{align} $$

Define

$$ \begin{align*} \eta_{o,g}(t) = \sum_{n=1}^\infty t^{-n} (\mathcal L_{\varphi}^n\mathbb 1_{[a] \times \{e\}})(o,g). \end{align*} $$

This converges for $t>\rho $ , but may either diverge or converge at $t=\rho $ . In the latter case, we need to modify the series so that it diverges at $t=\rho $ . We use the following standard lemma.

Let $b_n$ be the sequence given by the lemma corresponding to

$$ \begin{align*} a_n = (\mathcal L_{\varphi}^n\mathbb 1_{[a] \times \{e\}})(o,e) \end{align*} $$

and write $c_n=1/b_n$ . Let

$$ \begin{align*} \zeta_{o,g}(t) = \sum_{n=1}^\infty t^{-n} \frac{1}{c_n} (\mathcal L_{\varphi}^n\mathbb 1_{[a] \times \{e\}})(o,g). \end{align*} $$

If $o' \in [a]$ , then there is a natural bijection between the pre-images of o and $o'$ , so we can compare $\zeta _{o,g}(t)$ and $\zeta _{o',g}(t)$ term-by-term. Using Lemma 2.1, for all $g \in G$ and $t>\rho $ ,

(4.2) $$ \begin{align} e^{-B} \zeta_{o,g}(t) \le \zeta_{o',g}(t) \le e^B \zeta_{o,g}(t) \end{align} $$

and so the same sequence $c_n$ works for all $o \in [a]$ .

Lemma 4.2. Let $\rho '>\rho $ . For each $a \in S$ and each $g \in G$ ,

$$ \begin{align*} 0 < \inf_{o \in [a]} \inf_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)} \le \sup_{o \in [a]} \sup_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)} < \infty. \end{align*} $$

Proof. We begin by observing that, for every $g,h \in G$ , we have

$$ \begin{align*} (\mathcal L_{\varphi}^{n+k}\mathbb 1_{[a] \times \{e\}})(o,g) &\ge \sum_{\substack{\sigma^kz=o \\z_1=a\\ \psi_k(z)=h^{-1}g}} e^{\varphi^k(z)} \sum_{\substack{\sigma^nx=z \\x_1=a\\ \psi_n(x)= h}} e^{\varphi^n(x)} \\ &= \sum_{\substack{\sigma^kz=o \\z_1=a \\ \psi_k(z)=h^{-1}g}} e^{\varphi^k(z)} [(\mathcal L_\varphi^n\mathbb 1_{[a] \times \{e\}})(z,h)] \end{align*} $$

and (due to the transitivity of $T_\psi $ ) we can find a $k \ge 1$ so that the right-hand side is positive. Thus, we obtain

$$ \begin{align*} \zeta_{o,g}(t) &= \sum_{m=1}^k \frac{t^{-n}}{c_n} (\mathcal L_\varphi^m \mathbb 1_{[a] \times \{e\}})(o,g) + \sum_{n=1}^\infty \frac{t^{-(n+k)}}{c_{n+k}} (\mathcal L_{\varphi}^{n+k}\mathbb 1_{[a] \times \{e\}})(o,g) \\&\ge \sum_{m=1}^k \frac{t^{-n}}{c_n} (\mathcal L_\varphi^m \mathbb 1_{[a] \times \{e\}})(o,g) \\&\quad+ t^{-k}\sum_{\substack{\sigma^kz=o \\z_1=a\\ \psi_k(z)=h^{-1}g}} e^{\varphi^k(z)} \sum_{n=1}^\infty \frac{t^{-n}}{c_n} \frac{c_n}{c_{n+k}} [(\mathcal L_\varphi^n\mathbb 1_{[a] \times \{e\}})(z,h)]. \end{align*} $$

Since the numbers $c_n$ are positive and, for each fixed k, $\lim _{n \to \infty } c_n/c_{n+k}=1$ , we have $\inf _{n \ge 1} c_n/c_{n+k}>0$ . Furthermore,

$$ \begin{align*} \inf_{o \in [a]} \sum_{\substack{\sigma^kz=o \\z_1=a\\ \psi_k(z) = h^{-1}g}} e^{\varphi^k(z)}>0. \end{align*} $$

Hence, for $t \in (\rho ,\rho ']$ , we have

$$ \begin{align*} \zeta_{o,g}(t) \ge C_1(g,k,a) + \frac{C_2(g,h,k,a)}{C_3(k)}\zeta_{o,h}(t) \end{align*} $$

for positive $C_1,C_2$ and $C_3$ . We conclude that

$$ \begin{align*} \inf_{\rho <t\le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,h}(t)}>0. \end{align*} $$

Since g and h are arbitrary, we have

$$ \begin{align*} 0 < \inf_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)} \le \sup_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)} < \infty. \end{align*} $$

Using (4.2), the bounds can be made uniform over $o \in [a]$ .

Define

$$ \begin{align*} {\mathcal{Z}} = \bigcup_{n=0}^\infty \{z\in \Sigma \text{ : } \sigma^nz=o\} \end{align*} $$

the set of pre-images of o; this is a countable subset of $\Sigma $ .

Lemma 4.3. We can find a sequence $t_n \downarrow \rho $ such that the following limit exists and lies in $(0,\infty )$ for all $(z,g) \in \mathcal Z \times G$ :

$$ \begin{align*} \Phi(z,g) := \lim_{n \to \infty} \frac{\zeta_{z,g}(t_n)}{\zeta_{z,e}(t_n)}. \end{align*} $$

Furthermore, for all $g \in G$ ,

(4.3) $$ \begin{align} 0 < \inf_{z\in \mathcal Z \cap [a]} \Phi(z,g) \le \sup_{z\in \mathcal Z \cap [a]} \Phi(z,g) <\infty. \end{align} $$

Proof. Let $\{(z_m,g_m)\}_{m=1}^\infty $ be an enumeration of $\mathcal Z \times G$ and let $t_n^{(0)}$ be an arbitrary sequence in $(\rho ,\rho ']$ that is strictly decreasing to $\rho $ . Then, the numbers $\zeta _{z_1,g_1}(t_n^{(0)})/\zeta _{z_1,e}(t_n^{(0)})$ are a sequence in the compact interval

$$ \begin{align*} \bigg[\inf_{o \in [a]} \inf_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)}, \sup_{o \in [a]} \sup_{\rho < t \le \rho'} \frac{\zeta_{o,g}(t)}{\zeta_{o,e}(t)}\bigg]. \end{align*} $$

Hence, $t_n^{(0)}$ has a subsequence $t_n^{(1)}$ such that $\zeta _{z_1,g_1}(t_n^{(1)})/\zeta _{z_1,e}(t_n^{(1)})$ converges. Continue inductively to construct successive subsequences $t_n^{(m)}$ of $t_n^{(m-1)}$ such that $\zeta _{z_m,g_m}(t_n^{(m)})/\zeta _{z_m,e}(t_n^{(m)})$ converges. Now, take $t_n$ to be the diagonal sequence $t_n^{(n)}$ and convegence will hold for all $(z,g) \in \mathcal Z \times G$ . The inequalities (4.3) are immediate.

Lemma 4.4. For all $z \in \mathcal Z \cap [a]$ and all $n \ge 1$ ,

$$ \begin{align*} (\mathcal L_{\varphi}^n\Phi)(z,g) = \rho^n\Phi(z,g). \end{align*} $$

Proof. To simplify notation, we shall write the proof for the case $z=o$ and note that, using (4.2), the same argument holds for general $z \in \mathcal Z \cap [a]$ . Fix $k \ge 1$ and let $\epsilon>0$ . Since $\lim _{n \to \infty } c_{n-k}/c_n=1$ , we can choose $n_0$ such that $1-\epsilon \le c_{n-k}/c_n \le 1+\epsilon $ for all $n \ge n_0$ . Without loss of generality, we may take $n_0 \ge k$ . We will also use that

$$ \begin{align*} (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g) = \sum_{\sigma^k z=o} e^{\varphi^k(z)} \, (\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}). \end{align*} $$

Then, for $n \ge n_0$ ,

(4.4) $$ \begin{align} &\frac{1-\epsilon}{c_{n-k}} \sum_{\sigma^k z=o} e^{\varphi^k(z)} \, (\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}) \le \frac{1}{c_n} (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g) \nonumber \\& \quad\le \frac{1+\epsilon}{c_{n-k}} \sum_{\sigma^k z=o} e^{\varphi^k(z)} \, (\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}). \end{align} $$

Setting

$$ \begin{align*} C_1(g,t,n_0) = \sum_{n=1}^{n_0} \frac{t^{-n}}{c_n} (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g), \end{align*} $$

by (4.4), we have

$$ \begin{align*} &t^k \sum_{n=1}^\infty \frac{t^{-n}}{c_n} (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g)\\& \quad\le t^k C_1(g,t,n_0) + t^k(1+\epsilon) \sum_{\sigma^kz=o} e^{\varphi^k(z)} \, \sum_{n=n_0+1}^\infty \frac{t^{-n}}{c_{n-k}}(\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}). \end{align*} $$

Replacing t by the sequence $t_m$ from Lemma 4.3 and dividing by $\zeta _{z,e}(t_m)$ , we obtain

$$ \begin{align*} t_m^k \Phi(o,g) &\le \frac{t_m^kC_1(g,t_m,n_0)}{\zeta_{z,e}(t_m)} \\&\quad+ \frac{(1+\epsilon)}{\zeta_{z,e}(t_m)} \sum_{\sigma^kz=o} e^{\varphi^k(z)} \, \sum_{n=n_0+1}^\infty \frac{t_m^{-(n-k)}}{c_{n-k}}(\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}). \end{align*} $$

Then, letting $m \to \infty $ , so that $t_m \to \rho $ , this gives us

$$ \begin{align*} \rho^k \Phi(o,g) \le (1+\epsilon) \sum_{\sigma^kz=o} e^{\varphi^k(z)} \Phi(z, g\psi_k(z)^{-1}) \end{align*} $$

and, since $\epsilon $ is arbitrary,

$$ \begin{align*} \rho^k \Phi(o,g) \le \sum_{\sigma^kz=o} e^{\varphi^k(z)} \Phi(z, g\psi_k(z)^{-1}). \end{align*} $$

For the lower bound, we have

$$ \begin{align*} &t^k \sum_{n=1}^\infty \frac{t^{-n}}{c_n} (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g) \\&\quad\ge (1-\epsilon)\sum_{\sigma^k z=o} e^{\varphi^k(z)} \sum_{n=n_0+1}^\infty \frac{t^{-(n-k)}}{c_{n-k}} (\mathcal L_\varphi^{n-k} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}) \\&\quad= (1-\epsilon) \sum_{\sigma^k z=o} e^{\varphi^k(z)} \sum_{n=n_0-k+1}^\infty \frac{t^{-n}}{c_{n}} (\mathcal L_\varphi^{n} \mathbb 1_{[a] \times \{e\}})(z,g\psi_k(z)^{-1}), \end{align*} $$

which gives

$$ \begin{align*} \rho^k\Phi(o,g) \ge (\mathcal L_\varphi^k \Phi)(o,g).\\[-34pt] \end{align*} $$

Corollary 4.5. For all $(z,\gamma ) \in \mathcal Z \times G$ , we have

$$ \begin{align*} 0 < \inf_{g \in G} \frac{\Phi(z, g\gamma^{-1})}{\Phi(o,g)} \le \sup_{g \in G} \frac{\Phi(z,g\gamma^{-1})}{\Phi(o,g)} < \infty. \end{align*} $$

Proof. Here, we again use the topological transitivity of $T_\psi $ . Given $(z,\gamma ) \in \mathcal Z \times G$ , we can find $k \ge 1$ such that $\sigma ^kz=o'\in [a]$ and $\psi _k(z)=\gamma $ . This gives us

$$ \begin{align*} e^{\varphi^k(z)}\Phi(z,g\gamma^{-1}) &\le \sum_{\sigma^ky=o'} e^{\varphi^k(y)} \Phi(y,g\psi_k(y)^{-1}) \\ &=\rho^k\Phi(o',g) \le e^{2B} \rho^k \Phi(o,g). \end{align*} $$

Hence, $\sup _{g \in G} \Phi (z,g\gamma ^{-1})/\Phi (o,g)$ is finite.

Next, we find $x \in [a]$ and $k \ge 1$ such that $\sigma ^kx = z$ and $\psi _k(x)=\gamma ^{-1}$ . Then,

$$ \begin{align*} e^{-2B}\epsilon^{\varphi^k(x)}\Phi(o,g) &\le e^{\varphi^k(x)} \Phi(x,g) \\ &\le \sum_{\sigma^k y=o'} e^{\varphi^k(y)} \Phi(y,g\gamma^{-1}\psi_k(y)^{-1}) = \rho^k\Phi(z,g\gamma^{-1}). \end{align*} $$

Hence, $\inf _{g \in G} \Phi (z,g\gamma ^{-1})/\Phi (o,g)$ is positive.

We are now able to use the amenability of G to replace $\Phi $ with a function ${\phi : \mathcal Z \times G \to \mathbb R^{>0}}$ which, after normalization, is a homomorphism in the second factor. This will allow us to descend to $\overline {G}$ .

Lemma 4.6. There is a function $\phi : \mathcal Z \times G \to \mathbb R^{>0}$ such that:

1. (1) for all $g \in G$ and $n \ge 1$ ,

   $$ \begin{align*} (\mathcal L_{\varphi}^n \phi)(o,g) \le \rho^n \phi(o,g); \end{align*} $$

2. (2) for all $z \in \mathcal Z$ , the function

   $$ \begin{align*} \gamma \mapsto \frac{\phi(z,\gamma)}{\phi(z,e)} \end{align*} $$
   is a homomorphism from G to $\mathbb R^{>0}$ .

Furthermore, for all $a \in S$ and all $\gamma \in G$ , we have

$$ \begin{align*} \inf_{z \in \mathcal Z \cap [a]} \phi(z,\gamma)>0. \end{align*} $$

Proof. The construction of the function $\phi $ uses the amenability of G. Let M be a right-invariant Banach mean on $\ell ^\infty (G)$ . By Jensen’s inequality, if $\alpha $ is convex and ${f \in \ell ^\infty (G)}$ , then

$$ \begin{align*} M(\alpha(f)) \ge \alpha(M(f)). \end{align*} $$

We apply this to the functions

$$ \begin{align*} f_{z,\gamma}(g) = \log \frac{\Phi(z,g\gamma^{-1})}{\Phi(o,g)}, \end{align*} $$

where $(z,\gamma ) \in \mathcal Z \times G$ . It follows from Corollary 4.5 that $f_{z,\gamma } \in \ell ^\infty (G)$ for all ${(z,\gamma ) \in \mathcal Z \times G}$ . Thus, we have

$$ \begin{align*} M\bigg(g \mapsto \frac{\Phi(z,g\gamma^{-1})}{\Phi(o,g)}\bigg) = M\bigg(g \mapsto \exp \log \frac{\Phi(z,g\gamma^{-1})}{\Phi(o,g)}\bigg) \ge \exp M(f_{z,\gamma}). \end{align*} $$

Now, define $\phi : \mathcal Z \times G \to \mathbb R^{>0}$ by

$$ \begin{align*} \phi(z,\gamma) = \exp M(f_{z,\gamma}). \end{align*} $$

We will first prove that the inequality in part (i) holds. We need a little care as M is only finitely additive. Thus, we work with finite subsets of G that exhaust G. Let $\{g_k\}_{k=1}^\infty $ be an enumeration of G and, for any $N \ge 1$ , let $G_N = \{g_1,\ldots ,g_N\}$ . Then, Lemma 4.4 gives us

$$ \begin{align*} \rho^n &=M\bigg(g \mapsto \frac{(\mathcal L_\varphi^n \Phi)(o,g)}{\Phi(o,g)}\bigg) = M \bigg(g \mapsto \frac{1}{\Phi(o,g)}\sum_{\sigma^n z = o} e^{\varphi^n(z)} \Phi(z,g\psi_n(z)^{-1})\bigg) \\ &\ge M \bigg(g \mapsto \frac{1}{\Phi(o,g)}\sum_{\substack{\sigma^n z = o \\ \psi_n(z) \in G_N}} e^{\varphi^n(z)} \Phi(z,g\psi_n(z)^{-1})\bigg) \\ &= \sum_{\substack{\sigma^n z = o \\ \psi_n(z) \in G_N}} e^{\varphi^n(z)} M\bigg(g \mapsto \frac{\Phi(z,g\psi_n(z)^{-1})}{\Phi(o,g)}\bigg) \\ &\ge \sum_{\substack{\sigma^n z = o \\ \psi_n(z) \in G_N}} e^{\varphi^n(z)} \exp M\bigg(g \mapsto \log \frac{\Phi(z,g\psi_n(z)^{-1})}{\Phi(o,g)}\bigg) \\ &= \sum_{\substack{\sigma^n z = o \\ \psi_n(z) \in G_N}} e^{\varphi^n(z)} \phi(z,g\psi_n(z)^{-1}). \end{align*} $$

Taking the supremum over N gives the inequality in part (i).

Next, we show that part (ii) holds. First, note that we have

$$ \begin{align*} \log \phi(o,\gamma \delta) &= M\bigg(g \mapsto \log \frac{\Phi(o,g\delta^{-1}\gamma^{-1})}{\Phi(o,g)}\bigg) \\ &= M\bigg(g \mapsto \log \frac{\Phi(o,g\delta^{-1}\gamma^{-1})}{\Phi(o,g\delta^{-1})}\bigg) + M\bigg(g \mapsto \log \frac{\Phi(o,g\delta^{-1})}{\Phi(o,g)}\bigg) \\ &= \phi(o,\gamma) + \phi(o,\delta) \end{align*} $$

and

$$ \begin{align*} \log \phi(o,\gamma^{-1}) &= M\bigg(g \mapsto \log \frac{\Phi(o,g\gamma)}{\Phi(o,g)}\bigg) = M\bigg(g \mapsto \log \frac{\Phi(o,g)}{\Phi(o,g\gamma^{-1})}\bigg) \\ &= M\bigg(g \mapsto -\log \frac{\Phi(o,g\gamma^{-1})}{\Phi(o,g)}\bigg) = -\log \phi(o,\gamma). \end{align*} $$

Hence, $\gamma \mapsto \phi (o,\gamma )$ is a homomorphism. Next, we observe that

$$ \begin{align*} \log \phi(z,\gamma) &= M\bigg(g \mapsto \log \frac{\Phi(z,g\gamma)}{\Phi(o,g)}\bigg) \\ &= M\bigg(g \mapsto \log \frac{\Phi(o,g\gamma)}{\Phi(o,g)}\bigg) + M\bigg(g \mapsto \log \frac{\Phi(z,g\gamma)}{\Phi(o,g\gamma)}\bigg) \\ &= M\bigg(g \mapsto \log \frac{\Phi(o,g\gamma)}{\Phi(o,g)}\bigg) + M\bigg(g \mapsto \log \frac{\Phi(z,g)}{\Phi(o,g)}\bigg) \\ &= \log \phi(o,\gamma) + \log \phi(z,e). \end{align*} $$

Hence,

$$ \begin{align*} \gamma \mapsto \frac{\phi(z,\gamma)}{\phi(z,e)} = \phi(o,\gamma) \end{align*} $$

is a homomorphism.

To finish, we note that, by Corollary 4.5, $\phi (z,\gamma )>0$ and one easily sees that this positive lower bound can be taken uniformly in $z \in \mathcal Z \cap [a]$ .

Lemma 4.7. There is a function $\bar \phi : \mathcal Z \times \overline {G} \to \mathbb R^{>0}$ satisfying

$$ \begin{align*} (\mathfrak L_{\varphi}^n \bar \phi)(o,g)\le \rho^n \bar \phi(o,g) \end{align*} $$

for all $n \ge 1$ , and for all $a \in S$ and all $m \in \overline {G}$ , we have

$$ \begin{align*} \inf_{z \in \mathcal Z \cap [a]} \bar \phi(z,m)>0. \end{align*} $$

Proof. By part (ii) of Lemma 4.6, for each $z \in \mathcal Z$ , there is a homomorphism ${\Upsilon _z : \overline {G} \to \mathbb R^{>0}}$ such that

$$ \begin{align*} (\Upsilon_z \circ \pi)(\gamma) = \frac{\phi(z,\gamma)}{\phi(z,e)}. \end{align*} $$

We deduce that if $\pi (\gamma )=\pi (\delta )$ , then $\phi (z,\gamma )=\phi (z,\delta )$ , and we define $\bar \phi :\mathcal Z \times \overline {G} \to \mathbb R^{>0}$ by

$$ \begin{align*} \bar \phi(z,m) = \phi(z,\gamma) \end{align*} $$

for any $\gamma \in G$ satisfying $\pi (\gamma )=m$ .

We then have

$$ \begin{align*} (\mathfrak L_\varphi^n\bar\phi)(o,0) &= \sum_{\sigma^n z=o} e^{\varphi^n(z)} \bar\phi(z,\bar{\psi}_n(z)^{-1}) \\ &= \sum_{\sigma^n z=o} e^{\varphi^n(z)} \phi(z,\psi_n(z)^{-1}) \\ &= (\mathcal L_\varphi^n\phi)(o,e) \le \rho^n\phi(o,e) = \rho^n \bar\phi(o,0). \end{align*} $$

The final statement follows from Lemma 4.6 and the definition of $\bar \phi $ .

We now complete the proof of Theorem 1.3. Let

$$ \begin{align*} c = \inf_{z \in \mathcal Z \cap [a]} \bar\phi(z,0)>0. \end{align*} $$

Using Lemma 2.2, we have that

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_{\bar \psi}) &= \limsup_{n \to \infty}\frac{1}{n} \log (\mathfrak L_\varphi^n (c\mathbb 1_{[a] \times \{0\}}))(o,0) \\ &\le \limsup_{n \to \infty}\frac{1}{n} \log (\mathfrak L_\varphi^n \bar \phi)(o,0) \le \log \rho, \end{align*} $$

as required.

5 The BIP property and skew product extensions

We now introduce the stronger condition on $\Sigma $ required for the equidistribution result we described in §1. Let $\sigma : \Sigma \to \Sigma $ be a countable state Markov shift satisfying the BIP property. (The terminology BIP appears in, for example, the work of Sarig. If $\sigma : \Sigma \to \Sigma $ is topologically mixing, then BIP is equivalent to the finite primitivity condition used by Mauldin and Urbański [Reference Mauldin and Urbański21, Reference Mauldin and Urbański22]. Thus, we can cite results from both sets of authors.)

Let $\varphi : \Sigma \to \mathbb R$ be a locally Hölder continuous function. We say that a $\sigma $ -invariant probability measure $\mu $ on $\Sigma $ is a Gibbs measure for $\varphi $ if there are constants $A,B$ and P (depending only on $\varphi $ ) such that

$$ \begin{align*} A \le \frac{\mu([w])}{e^{\varphi^n(x)-nP}} \le B \end{align*} $$

for all $w \in \mathcal W_n$ , for all $n \ge 1$ and all $x \in [w]$ . If $\varphi $ has a Gibbs measure, then $P=P_{\mathrm {G}}(\varphi ,\sigma )$ .

From now on, we assume that $\sigma : \Sigma \to \Sigma $ is topologically mixing and satisfies BIP. Let $\varphi : \Sigma \to \mathbb R$ be locally Hölder continuous and summable, so that $P_{\mathrm {G}}(\varphi ,\sigma )$ is finite. Then, $\varphi $ has a unique Gibbs measure, which we denote by $\mu _\varphi $ , and $P=P_{\mathrm {G}}(\varphi )$ (see [Reference Sarig30, Theorem 1]). We also have a characterization of the Gurevič pressure in terms of periodic points:

(5.1) $$ \begin{align} P_{\mathrm{G}}(\varphi,\sigma) = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\sigma^nx=x} e^{\varphi^n(x)} \end{align} $$

(see [Reference Sarig30, Corollary 1]). (Note that the equality (5.1) may fail in the absence of the BIP property.)

In the presence of BIP, we also have stronger results for our transfer operators acting on an appropriate Banach space. Let $\mathcal F_\theta ^b$ denote the space of bounded $\theta $ -locally Hölder continuous functions $v: \Sigma \to \mathbb {C}$ with the norm $\|\cdot \|_\theta = \|\cdot \|_\infty +|\cdot |_\theta $ , where

$$ \begin{align*} |v|_\theta= \sup_{n \ge 1} \sup_{w \in \mathcal W_n} \sup_{x,y \in [w]} \frac{|v(x)-v(y)|}{\theta^n}. \end{align*} $$

If $\varphi : \Sigma \to \mathbb R$ is summable, then $L_\varphi : \mathcal F_\theta ^b \to \mathcal F_\theta ^b$ is a well-defined bounded linear operator [Reference Mauldin and Urbański22, Lemma 2.4.1].

We have the following analyticity result.

Lemma 5.1. [Reference Mauldin and Urbański22, Theorem 2.6.13 and Proposition 2.6.13]

Let $\sigma : \Sigma \to \Sigma $ be topologically mixing and satisfy BIP, and let $\varphi ,\theta : \Sigma \to \mathbb R$ be weakly Hölder continuous. Suppose there exists $\epsilon>0$ such that $\varphi +t\theta $ is summable for $|t|<\epsilon $ . Then, $t \mapsto P_{\mathrm {G}}(\varphi +t\theta ,\sigma )$ is real analytic on $(-\epsilon ,\epsilon )$ and

$$ \begin{align*} \frac{dP_{\mathrm{G}}(\varphi+t\theta,\sigma)}{dt} \bigg|_{t=0} =\int \theta\, d\mu_\varphi. \end{align*} $$

We need an analogue of (5.1) for $T_\psi $ . If $T_\psi : \Sigma \times G \to \Sigma \times G$ is mixing, then we have the following lemma.

Lemma 5.2. Let $\sigma : \Sigma \to \Sigma $ be topologically mixing and satisfy BIP, and furthermore, let $T_\psi : \Sigma \times G \to \Sigma \times G$ be topologically mixing. Then, we have

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_\psi) = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\substack{\sigma^nx=x \\\psi_n(x)=e}} e^{\varphi^n(x)}. \end{align*} $$

Proof. It is immediate from the definition of Gurevič pressure that

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_\psi) \le \lim_{n \to \infty} \frac{1}{n} \log \sum_{\substack{\sigma^nx=x \\\psi_n(x)=e}} e^{\varphi^n(x)}. \end{align*} $$

For the other direction, we adapt the proof of [Reference Sarig30, Theorem 1]. Given $w \in \mathcal W_n$ , let $ \varphi ^n_+(w) = \sup \{\varphi ^n(y) \text { : } y \in [w]\}$ . Now, fix $a \in S$ . By the mixing of $T_\psi $ , there exists ${n_0 \ge 1}$ and, for $i=1,\ldots ,N$ , $w_{a,b_i}, w_{b_i,a} \in \mathcal W_{n_0}$ such that $aw_{a,b_i}b_i \in \mathcal W_{n_0 +2}$ , $b_iw_{b_i,a} a \in \mathcal W_{n_0+2}$ , $\psi _{n_0}(w_{a,b_i})=e$ and $\psi _{n_0}(w_{b_i,a})=e$ . By the BIP property, given $w \in \mathcal W_n$ , there exists $i,j \in \{i,\ldots ,N\}$ such that $b_iwb_j \in \mathcal W_{n+2}$ . Hence,

$$ \begin{align*} aw_{a,b_i} b_i w b_j w_{b_j,a} a \in \mathcal W_{n+M+1}, \end{align*} $$

where $M = 2n_0+3$ , and concatenating $aw_{a,b_i} b_i w b_j w_{b_j,a}$ gives a periodic point of period $n+M$ . Furthermore, if $\psi _n(w)=e$ , then

$$ \begin{align*} |\psi_{n+M}(aw_{a,b_i} b_i w b_j w_{b_j,a})| \end{align*} $$

is bounded independently of n. We conclude that there exists $n_1 \ge 1$ and $C>0$ so that

(5.2) $$ \begin{align} \sum_{|g|\le n_1} \sum_{\substack{\sigma^{n+M}x=x \\ \psi_{n+M}(x)=g}} e^{\varphi^{n+M}(x)} \ge C \sum_{\substack{w \in \mathcal W_n \\ \psi_n(w)=e}} e^{\varphi^n_+(w)} \ge C (\mathcal L_\varphi^n \mathbb 1_{[a] \times \{e\}})(o,g) \end{align} $$

for any choice of $o \in [a]$ .

Now, the mixing of $T_\psi $ implies that

$$ \begin{align*} \lim_{n \to \infty} \frac{1}{n} \log \sum_{\substack{\sigma^{n}x=x \\ \psi_{n}(x)=g}} e^{\varphi^{n}(x)} \end{align*} $$

is independent of $g \in G$ . Combining this with (5.2), we see that

$$ \begin{align*} \lim_{n \to \infty} \frac{1}{n} \log \sum_{\substack{\sigma^nx=x \\\psi_n(x)=e}} e^{\varphi^n(x)} \ge P_{\mathrm{G}}(\varphi,T_\psi), \end{align*} $$

as required.

We now suppose that $\varphi $ and $\psi $ satisfying Assumptions I, II and III, stated in §1. To simplify notation, we identify $\bar \psi : \Sigma \to \overline {G}$ with a function ${f : \Sigma \to \mathbb Z^d}$ and we now denote the abelian skew product by $T_f$ .

Assumption (I) immediately implies that $T_f\! : \Sigma \times \mathbb Z^d \to \Sigma \times \mathbb Z^d$ is also topologically mixing. Furthermore, Assumption (I) implies that f is aperiodic: if $e^{2\pi i \langle t,f(x)\rangle } = z u(\sigma x)/u(x)$ for all $x {\kern-1pt}\in{\kern-1pt} \Sigma $ , where $t {\kern-1pt}\in{\kern-1pt} \mathbb R^d$ , $z {\kern-1pt}\in{\kern-1pt} \mathrm {U}(1) {\kern-1pt}={\kern-1pt} \{z \in \mathbb {C} \text { : }|z|{\kern-1pt}={\kern-1pt}1\}$ and ${u {\kern-1pt}\in{\kern-1pt} C(\Sigma ,\mathrm {U}(1))}$ , then $t=0\in \mathbb R^d/\mathbb Z^d$ and $z=1$ . If Assumption (II) holds, then we easily see that $\varphi +\langle w,f \rangle $ is summable, $L_{\varphi + \langle w,f \rangle } : \mathcal F_\theta ^b \to \mathcal F_\theta ^b$ is a well-defined bounded linear operator and

$$ \begin{align*} \mathfrak p(w):=P_{\mathrm{G}}(\varphi+\langle w,f \rangle,\sigma) \end{align*} $$

is finite for $|w|<\delta (\varphi ,f)$ .

We also need to consider the transfer operator with complex potentials. As in §1, $\delta =\delta (\varphi ,f)$ and we write

$$ \begin{align*} B_{\mathbb{C}}(\delta) = \{s \in \mathbb{C}^d \text{ : } |\mathrm{Re}(s)|<\delta\}. \end{align*} $$

One can easily check that $s \mapsto L_{\varphi +\langle s,f\rangle }$ is analytic on $B_{\mathbb {C}}(\delta )$ (cf. [Reference Mauldin and Urbański22, Corollary 2.6.10]). Furthermore, for s in a neighbourhood of $B(\delta )$ in $B_{\mathbb {C}}(\delta )$ , $L_{\varphi +\langle s,f\rangle }$ has a simple isolated maximal eigenvalue $\unicode{x3bb} (s)$ such that

$$ \begin{align*} \unicode{x3bb}(w) = e^{\mathfrak p(w)} \end{align*} $$

for $w \in B(\delta )$ [Reference Kesseböhmer and Kombrink18, Proposition 2.8]). As a consequence, using spectral perturbation theory, we have that $s \mapsto e^{\mathfrak p(s)}$ is analytic for s is a neighbourhood of $B(\delta )$ in $B_{\mathbb {C}}(\delta )$ .

We may also calculate the derivatives of $\mathfrak p$ . From Lemma 5.1, we then have

$$ \begin{align*} \nabla \mathfrak p(w)= \int f \, d\mu^{w}, \end{align*} $$

where

$$ \begin{align*} \mu^w := \mu_{\varphi + \langle w.f \rangle}. \end{align*} $$

Furthermore, the function $\mathfrak p$ is strictly convex. (Strict convexity fails only if unless there is $w \ne 0$ such that $\langle w,f \rangle $ is cohomologous to a constant, which violates the transitivity of $T_f$ .)

Under Assumption (III), we have

$$ \begin{align*} \nabla \mathfrak p(\xi)= \int f \, d\mu^\xi =0. \end{align*} $$

The value $\xi $ is important and allows us to relate the Gurevič pressure $P_{\mathrm {G}}(\varphi ,T_f)$ to a Gurevič pressure with respect to $\sigma $ . First, we give a technical lemma. We observe that since f is valued in $\mathbb Z^d$ , $L_{\varphi +\langle \xi +2\pi it,f\rangle }$ is well defined for $t \in \mathbb R^d/\mathbb Z^d$ .

Proposition 5.4. We have

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_f) = P_{\mathrm{G}}(\varphi+\langle \xi,f\rangle,\sigma). \end{align*} $$

Proof. By the definition of Gurevič pressure (and recalling our convention that ${\varphi : \Sigma \times \mathbb Z^d \to \mathbb R}$ is defined by $\varphi (x,\cdot )=\varphi (x)$ ), we have

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_f) = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\substack{\sigma^nx=x \\ f^n(x)=0}} e^{\varphi^n(x)} \mathbb{1}_{[a]}(x). \end{align*} $$

However, we may make the following calculation. Choose $o \in [a]$ and set

$$ \begin{align*} \mathcal Z(n,a,o) = \{y \in [a] \text{ : } \sigma^ny=o\}. \end{align*} $$

There is a natural bijection $q : \mathrm {Fix}_n(\sigma ) \cap [a] \to \mathcal Z(n.a,o)$ given by $q(x) = x_1\cdots x_no$ . We have

$$ \begin{align*} \sum_{\substack{\sigma^nx=x \\ f^n(x)=0x}} e^{\varphi^n(x)} \mathbb{1}_{[a]}(x) &=\sum_{\substack{\sigma^nx=x \\ f^n(x)=0}} e^{\varphi^n(x)+\langle \xi,f^n(x)\rangle} \mathbb{1}_{[a]}(x) \\ &= \int_{\mathbb R^d/\mathbb Z^d} \sum_{x \in \mathrm{Fix}_n(\sigma) \cap [a]} e^{\varphi^n(x)+\langle \xi+2\pi it,f^n(x)\rangle} \, dt \\ &= \int_{\mathbb R^d/\mathbb Z^d} \sum_{y \in \mathcal Z(n,a,z)} e^{\varphi^n(y)+\langle \xi+2\pi it,f^n(y)\rangle} \, dt + O(\theta^n e^{n\mathfrak p(\xi)}) \\ &= \int_{\mathbb R^d/\mathbb Z^d} (L^n_{\varphi+\langle \xi+2\pi i t,f\rangle} \mathbb{1}_{[a]})(z) + O(\theta^n e^{n\mathfrak p(\xi)}) \end{align*} $$

for some $0<\theta <1$ . Using Lemma 5.3 and the analysis in [Reference Pollicott and Sharp27], we can show that the right-hand side behaves like $e^{n\mathfrak p(\xi )} n^{-d/2}$ , so the result follows.

6 Equidistribution for skew product extensions

In this section, we give the proof of Theorem 1.1. We give the proof for the sequence $\mathfrak M_\varphi (\Lambda ^\psi (n))$ , the other cases being similar. The proof is based on the following lemma.

Lemma 6.1. Let $g : \Sigma \to \mathbb R$ be a bounded locally Hölder continuous function. Given ${\epsilon>0}$ , there exists $C>0$ and $\eta>0$ such that

$$ \begin{align*} \frac{1}{\Pi_\varphi(\Lambda^\psi(n))} \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int g \, d\tau_{x,n} -\int g \, d\mu_\xi\!|>\epsilon}} e^{\varphi^n(x)} \le Ce^{-\eta n}. \end{align*} $$

Proof. We consider $x \in \Lambda ^\psi (n)$ such that $\int g \, d\tau _{x,n}> \int g \, d\mu _\xi +\epsilon $ and $\int g \, d\tau _{x,n} < \int g \, d\mu _\xi -\epsilon $ separately. For $s>0$ , we have

$$ \begin{align*} \sum_{\substack{x \in \Lambda^\psi(n) \\ \int g \, d\tau_{x,n}>\int g \, d\mu^\xi +\epsilon}} e^{\varphi^n(x)} \le \sum_{x \in \Lambda^\psi(n)} e^{\varphi^n(x) + sg^n(x) - ns\int g \, d\mu^\xi -ns\epsilon}, \end{align*} $$

so that

$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\substack{x \in \Lambda^\psi(n) \\ \int g \, d\tau_{x,n}>\int g \, d\mu^xi +\epsilon}} e^{\varphi^n(x)} \le P_{\mathrm{G}}(\varphi+\langle \xi,f\rangle +sg,\sigma) -s\int g \, d\mu^\xi -s\epsilon. \end{align*} $$

Similarly, for $s<0$ , we have

$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\substack{x \in \Lambda^\psi(n) \\ \int g \, d\tau_{x,n} <\int g \, d\mu^\xi -\epsilon}} e^{\varphi^n(x)} \le P_{\mathrm{G}}(\varphi+\langle \xi,f\rangle +sg,\sigma) -s\int g \, d\mu^\xi +s\epsilon. \end{align*} $$

If we write $\mathfrak p_g(s) := P_{\mathrm {G}}(\varphi +\langle \xi ,f\rangle +sg)$ then, applying Lemma 5.1, we have a bound of the form

$$ \begin{align*} \limsup_{n \to \infty} \frac{1}{n} \log \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int g \, d\tau_{x,n} -\int g \, d\mu_\xi\!|>\epsilon}} e^{\varphi^n(x)} \le \mathfrak p_g(s) - s\mathfrak p_g'(0) -\epsilon |s|, \end{align*} $$

which is strictly smaller than $P_{\mathrm {G}}(\varphi +\langle \xi ,f \rangle )$ for sufficiently small values of s. The proof is completed by noting that, by Theorem 1.3 and Proposition 5.4,

$$ \begin{align*} \lim_{n \to \infty} \frac{1}{n} \log \Pi_\varphi(\Lambda^\psi(n)) = P_{\mathrm{G}}(\varphi+\langle \xi,f \rangle,\sigma). \\[-39pt] \end{align*} $$

Proof of Theorem 1.1.

Write $\mathfrak M_n = \mathfrak M_\varphi (\Lambda ^\psi (n))$ . We need to show that for every bounded continuous function $g : \Sigma \to \mathbb R$ , we have

$$ \begin{align*} \lim_{n \to \infty} \int g \, d\mathfrak M_n = \int g \, d\mu^\xi. \end{align*} $$

It is enough to prove this when g is a bounded locally Hölder continuous function. We have

$$ \begin{align*} &\int g \, d\mathfrak M_n - \int g \, d\mu^\xi\\& \quad = \frac{1}{\Pi_\varphi(\Lambda^\psi(n))} \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int \chi \, d\tau_{x,n} - \int \chi d\mu^\xi\!|> \epsilon}} e^{\varphi^n(x)} \int g \, d\tau_{x,n} + \frac{1}{\Pi_\varphi(\Lambda^\psi(n))}\\& \qquad \times \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int \chi \, d\tau_{x,n} - \int g\,d\mu^\xi\!| \le \epsilon}} \times\ e^{\varphi^n(x)} \!\int\! g \, d\tau_{x,n}-\!\int\!\! \chi \, d\mu^\xi. \end{align*} $$

By Lemma 6.1, the first term on the right-hand side tends to zero exponentially fast. Also,

$$ \begin{align*} &\bigg| \frac{1}{\Pi_\varphi(\Lambda^\psi(n))} \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int g \, d\tau_{x,n} - \int g\,d\mu^\xi\!| \le \epsilon}} e^{\varphi^n(x)} \int g \, d\tau_{x,n}-\int g \, d\mu^\xi \bigg| \\ &\quad\le \epsilon + \frac{|\int g \, d\mu^\xi|}{\Pi_\varphi(\Lambda^\psi(n))} \sum_{\substack{x \in \Lambda^\psi(n) \\ |\!\int g \, d\tau_{x,n} - \int g\,d\mu^\xi\!|> \epsilon}} e^{\varphi^n(x)} \le \epsilon + Ce^{-\eta(\epsilon)n}, \end{align*} $$

which, since $\epsilon $ is arbitrary, gives the result.

7 Equality implies amenability

In this section, we show that, under Assumptions (I), (II) and (III), the converse of Theorem 1.3 holds, that is, if $P_{\mathrm {G}}(\varphi ,T_\psi )=P_{\mathrm {G}}(\varphi ,T_{\bar \psi })$ , then G is amenable. As above, we write $f = \bar {\psi }$ .

To do this, we need to specify a function space for $\mathcal L_\varphi $ to act on. Let $\mathcal H$ denote the set of continuous functions $v : \Sigma \times G \to \mathbb {C}$ such that $g \mapsto \|v(\cdot ,g)\|_\infty $ is in $\ell ^2(G)$ , with norm

$$ \begin{align*} \|v\|_{\mathcal H} = \bigg(\sum_{g \in G} \|v(\cdot,g)\|_\infty^2\bigg)^{1/2}, \end{align*} $$

and let $\mathrm {spr}_{\mathcal H}(\mathcal L_\varphi )$ denote the spectral radius of $\mathcal L_\varphi $ on $\mathcal H$ . We always have

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_\psi) \le \log \mathrm{spr}_{\mathcal H}(\mathcal L_{\varphi}) \le P_{\mathrm{G}}(\varphi,\sigma). \end{align*} $$

We state a result due to Stadlbauer.

We can now prove the desired result.

Proof of Proposition 1.5.

Let $\xi \in \mathbb R^d$ be as in the previous section and let $\varphi _\xi = \varphi +\langle \xi ,f \rangle $ . Suppose that G is not amenable. Then,

$$ \begin{align*} P_{\mathrm{G}}(\varphi,T_\psi) = P_{\mathrm{G}}(\varphi_\xi,T_\psi) \le \log \mathrm{spr}_{\mathcal H}(\mathcal L_{\varphi_\xi}) < P_{\mathrm{G}}(\varphi_\xi,\sigma) = P_{\mathrm{G}}(\varphi,T_{\bar \psi}), \end{align*} $$

where the strict inequality uses Proposition 7.1.