that $\bar v=0$. Then for any $\epsilon>0$, there exists $\theta\in(0,1)$, $m\ge1$, such that $$\begin{aligned}
    \rho(t)=O\bigl(\|v\|_\theta|w|_{\infty,m}t^{-(\beta-\epsilon)}\bigr),\end{aligned}$$ for all $v\in F_\theta(\tilde Y)$, $w\in L^{\infty,m}(\tilde Y)$, $t>0$.

It is well-known that the regular variation assumption is necessary for Theorem \[thm-infinite\]. The assumption of polynomial tails in Theorem \[thm-finite\] can be relaxed as in [@Gouezel-phd] or [@MTapp] but we do not pursue that here.

Examples with Gibbs-Markov first return maps {#sec-GM}
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In formulating Theorems \[thm-infinite\] and \[thm-finite\], we considered suspensions where the map $F:Y\to Y$ is uniformly expanding and the roof function $\varphi:Y\to{{\mathbb R}}^+$ is unbounded. Often it is convenient to reverse the roles and to start with a map $f:X\to X$ that is less well-behaved (nonuniformly expanding instead of uniformly expanding) together with a bounded roof function $\varphi_X:X\to{{\mathbb R}}$.

In particular a large class of examples covered by our methods are those where the map $f:X\to X$ has a first return map $F:Y\to Y$ that is Gibbs-Markov and where $\varphi_X$ is globally Lipschitz. This includes suspensions of parabolic rational maps of the complex plane (Aaronson [*et al*]{} [@ADU93]) and Thaler’s class of interval maps with indifferent fixed points [@Thaler95] (in particular the family  defined above).

The separation time $s$, and hence the 