continuous time system. In contrast, the uniformly expanding Poincaré map $F$ is much more tractable.

Hence, at least for certain situations in dynamical systems theory, and perhaps in probability theory too, the renewal equation presented here seems a more useful approach then passing to discrete time skeletons.

The following elementary result is required in both Part \[part-infinite\] and Part \[part-finite\].

\[prop-m\] Let $m\ge1$. Suppose that $v\in L^1(\tilde Y)$ and $w\in L^{\infty,m}(\tilde Y)$. Then $$\hat\rho_{v,w}(s)=
\sum_{j=1}^m \rho_{v,\partial_t^{j-1}w}(0)s^{-j}+s^{-m}\hat \rho_{v,\partial_t^mw}(s).$$

First note that $\rho_{v,w}$ is $m$-times differentiable and ${\rho_{v,w}}^{(j)}=\rho_{v,\partial_t^jw}$ for $j=0,\dots,m$. By Taylor’s Theorem, $\rho_{v,w}(t)=P_m(t)+H_m(t)$, where $$P_m(t)=\sum_{j=0}^{m-1} \frac{1}{j!}{\rho_{v,w}}\!^{(j)}(0)t^j, \quad
H_m(t)=\int_0^t g(t-\tau){\rho_{v,w}}\!^{(m)}(\tau)\,d\tau,
\quad g(t)=\frac{t^{m-1}}{(m-1)!}.$$ Hence $\hat\rho_{v,w}(s)=\sum_{j=0}^{m-1}\rho_{v,\partial_t^jw}(0)s^{-(j+1)}+
\hat H_m(s)$, where $\hat H_m(s)=
\hat g(s)\hat \rho_{v,\partial_t^mw}(s)
=s^{-m}\hat \rho_{v,\partial_t^mw}(s)$.

Dolgopyat-type estimates
========================

In this section, we recall estimates of [@Dolgopyat98b] extended to the nonuniformly hyperbolic setting [@M07; @M09]. These are required to control $(I-\hat R(s))^{-1}$ for $s=ib$, $b$ large. The arguments need some modification here to allow for the possibility that $\varphi\not\in L^1$.

We recall that the twisted transfer operators $\hat R_0(s):L^1(Y)\to L^1(Y)$ satisfy for $n\ge1$, $$\begin{aligned}
 \label{eq-Rn}
(\hat R_0(s)^nv)(y)=\sum_{a\in\alpha_n}e^{p_n(y_a)}e^{-s\varphi_n(y_a)}v(y_a),\end{aligned}$$ where $y_a$ denotes the unique preimage $y_a\in a\cap F^{-n}y$ and $\varphi_n=\sum_{j=0}^{n-1}\varphi\circ F^j$.

\[lem-Rb\] For every $\epsilon>0$, there exist constants $C\ge1$ and $\theta,\tau\in(0,1)$ such that for every $v\in 