orbit for $f_t$ is given by $\tau=\sum_{j=0}^{k-1}\varphi(F^jy)$ and so $v(y)=e^{ib\tau}v(y)$. Dividing by $v(y)$, we obtain $e^{ib\tau}=1$ verifying the claim.

Let $\hat T_0(s)=(I-\hat R_0(s))^{-1}$; this is well-defined for $s\in\overline{{\mathbb H}}-\{0\}$. Write $v^u(y)=v(y,u)$. Then $$((I-\hat R(s))^{-1}v)(y,u)=(\hat T_0(s)v^u)(y),\quad s\in\overline{{\mathbb H}}-\{0\}.$$ Also we obtain the renewal equation $$\hat T(s)=\hat U(s)(I-\hat R(s))^{-1}, \quad s\in\overline{{\mathbb H}}-\{0\}.$$ By Proposition \[prop-renew\], we obtain an analytic extension $$\begin{aligned}
 \label{eq-rhohat}
\hat\rho(s)=\int_{\tilde Y}\hat U(s)(I-\hat R(s))^{-1}v\,w\,d\mu^\varphi,\end{aligned}$$ defined on a neighbourhood of $\overline{{\mathbb H}}-\{0\}$.

\[rmk-oprenew\] The operator renewal equation $\hat T(s)=\hat U(s)(I-\hat R(s))^{-1}$ has the desired effect of relating the Laplace transform of the transfer operators $T_t$ for the flow with the perturbed transfer operator $\hat R_0(s)v=R(e^{-s\varphi}v)$ where $R$ is the transfer operator for the Poincaré map $F$.

In dynamical systems theory, there are two standard types of discrete time system that can be obtained from a continuous time system: Poincaré maps such as $F$ and the time-$h$ map $f_h$ for fixed $h>0$. In the probability theory literature, a standard technique after Kingman [@Kingman63] is to consider discrete time “skeletons” $f_h$ and to pass to the continuous time limit as $h\to0$. However, for the properties studied in the current paper, the partially hyperbolic time-$h$ map is as difficult to study as the underlying 