assumed that the partition $\alpha$ separates orbits of $F$, so $s(y,y')$ is finite for all $y\neq y'$. Then $d_\theta$ is a metric. Let $F_\theta(Y)$ be the Banach space of $d_\theta$-Lipschitz functions $v:Y\to{{\mathbb R}}$ with norm $\|v\|=|v|_\infty+|v|_\theta$ where $|v|_\theta$ is the Lipschitz constant of $v$.

Define the potential function $p=\log\frac{d\mu}{d\mu\circ F}:Y\to{{\mathbb R}}$. We require that $p$ is uniformly piecewise Lipschitz: that is, $p|_a$ is $d_\theta$-Lipschitz for each $a\in\alpha$ and the Lipschitz constants can be chosen independent of $a$. We also require the big images property $\inf_a \mu(Fa)>0$. A [*Gibbs-Markov map*]{} is a Markov map with uniformly piecewise Lipschitz potential and satisfying the big images property.

For $n\ge1$ we let $\alpha_n$ denote the partition into $n$-cylinders. Let $p_n=\sum_{j=0}^{n-1}p\circ F^j$. It follows from the Lipschitz property of $p$ together with big images that there exists a constant $C_1>0$ such that $$\begin{aligned}
 \label{eq-GM}
e^{p_n(y)}\le C_1\mu(a), \quad\text{and}\quad
|e^{p_n(y)}-e^{p_n(y')}|\le C_1\mu(a)d_\theta(F^ny,F^ny'),\end{aligned}$$ for all $y,y'\in a$, $a\in\alpha_n$, $n\ge1$.

#### Roof function

The roof function $\varphi:Y\to{{\mathbb Z}}^+$ is assumed to be piecewise Lipschitz with respect to $d_{\theta_0}$ for some $\theta_0\in(0,1)$, (ie $1_a\varphi\in F_{\theta_0}(Y)$ for all $a\in\alpha$), and satisfying ${\operatorname{ess\, inf}}\varphi>0$. For convenience of notation, we suppose that ${\operatorname{ess\, inf}}\varphi>2$. In particular, the set $\tilde Y=Y\times[0,1]$ lies inside $Y^\varphi$.

We 