$n\ge1$. Three positive constants arise frequently throughout the paper: $C_1$ and $C_2$ introduced in Section \[sec-main\], and $c_2=(C_2+1)^{-1}$ which appears for the first time in Section \[sec-decomp\].

Statement of the main results {#sec-main}
=============================

#### Suspension semiflows

Let $(Y,\mu)$ be a probability space and $F:Y\to Y$ an ergodic measure-preserving transformation. Let $\varphi:Y\to{{\mathbb R}}^+$ be a measurable roof function. Form the suspension $Y^\varphi=\{(y,u)\in Y\times{{\mathbb R}}: 0\le u\le \varphi(y)\}/\sim$ where $(y,\varphi(y))\sim(Fy,0)$. The suspension flow $f_t:Y^\varphi\to Y^\varphi$ is given by $f_t(y,u)=(y,u+t)$ computed modulo identifications and the measure $\mu^\varphi=\mu\times{\rm Lebesgue}$ is ergodic and $f_t$-invariant. In the finite measure case, we normalise by $\bar\varphi=\int_Y\varphi\,d\mu$ so that $\mu^\varphi=(\mu\times{\rm Lebesgue})/{\bar\varphi}$ is a probability measure.

#### Gibbs-Markov maps

We assume throughout that $F:Y\to Y$ is Gibbs-Markov. Roughly speaking $F$ is uniformly expanding with big images and good distortion properties.

We recall the key definitions [@Aaronson]. Let $(Y,\mu)$ be a Lebesgue probability space with countable measurable partition $\alpha$. Let $F:Y\to Y$ be an ergodic, conservative, measure-preserving, Markov map transforming each partition element bijectively onto a union of partition elements. Fix $\theta\in(0,1)$ and define $d_\theta(y,y')=\theta^{s(y,y')}$ where the [*separation time*]{} $s(y,y')$ is the greatest integer $n\ge0$ such that $F^ny$ and $F^ny'$ lie in the same partition element in $\alpha$. It is 