$f:X\to X$ is an intermittent map of the form , with $\alpha\in(0,2)$. Let $\varphi_X:X\to{{\mathbb R}}^+$ be a $C^\eta$-roof function, $\eta\in(0,1]$. Suppose further that the suspension semiflow possesses a pair of periodic orbits with Diophantine ratio of periods. Let $\beta=1/\alpha$ and $c_0=\frac14\beta^\beta\varphi_X(0)^\beta h(\frac12)$. Then

-   For $\alpha\in[1,2)$, the conclusion of Theorem \[thm-infinite\](a) holds with $\tilde\ell(t)\sim c_0$ for $\alpha\in(1,2)$ and $\tilde\ell(t)\sim c_0\log t$ for $\alpha=1$.

    Moreover if $\eta\in(0,1]$ is sufficiently large ($\eta\in(\frac{1-\beta}{\beta},1]$ suffices), then the conclusion of Theorem \[thm-infinite\](b) holds.

-   For $\alpha\in(0,1)$, the conclusion of Theorem \[thm-finite\](a) holds in the form $$\rho_{v,w}(t)-\bar v\bar w =(1/{\bar\varphi})c_0(\beta-1)^{-1}\bar v\bar w\, t^{-(\beta-1)}
    +O(t^{-p}),$$ where $p=\min\{\beta-1+\beta\eta,2\beta-2\}$ for $\beta<2$ and $p=\min\{\beta-1+\beta\eta,\beta-\epsilon\}$ for $\beta\ge2$.

    Moreover, if $\bar v=0$, then we obtain $\rho_{v,w}(t)=O(t^{-(\beta-\epsilon)})$ as in Theorem \[thm-finite\](b).

Condition (A1) is automatic, and we have explicitly assumed condition (A2). Proposition \[prop-PMphi\] gives the required estimates on $\mu(\varphi>t)$. Hence the results follow from Theorems \[thm-infinite\] and \[thm-finite\].

\[part-oprenew\]

In this part of the paper, we formulate an operator renewal equation for flows.

The operator renewal equation {#sec-renew}
=============================

#### Transfer operators

Let $R:L^1(Y)\to L^1(Y)$ denote the transfer operator for $F:Y\to Y$ and let $L_t:L^1(Y^\varphi)\to L^1(Y^\varphi)$ denote the family of transfer operators for $f_t$. (So $\int_Y Rv\,w\,d\mu=\int_Y v\,w\circ F\,d\mu$, $\int_{Y^\varphi} L_tv\,w\,d\mu^\varphi=\int_{Y^\varphi} v\,w\circ 