metric $d_\theta$, extends from $Y$ to $X$: define $s(f^\ell y,f^\ell y')=s(y,y')$ for all $y,y'\in a$, $a\in\alpha$, $0\le\ell<\tau(y)$. Suppose that the roof function $\varphi_X:X\to{{\mathbb R}}^+$ is locally Lipschitz with respect to this metric and define the induced roof function $\varphi:Y\to{{\mathbb Z}}^+$, $\varphi(y)=\sum_{j=0}^{\tau(y)-1}\varphi_X(y)$. Thus we obtain equivalent semiflows using either $(f,\varphi_X)$ or $(F,\varphi)$. Furthermore, condition (A1) is automatic if $\varphi_X$ is globally Lipschitz, and the statement of condition (A2) is unchanged in the new setting (the set of periods for $(f,\varphi_X)$ or $(F,\varphi)$ are identical).

\[ex-PM\] Consider the family of intermittent maps  discussed in the introduction. Such maps $f:X\to X$ have a Gibbs-Markov first return map to the set $Y=[\frac12,1]$.

\[prop-PMphi\] Suppose that $\varphi_X:X\to{{\mathbb R}}^+$ is $C^\eta$, $\eta\in(0,1]$. Then $$\mu(y\in Y:\varphi(y)>t)=c_0 t^{-\beta}(1+O(t^{-\beta\eta})),$$ where $\beta=1/\alpha$, $c_0=\frac14\beta^\beta\varphi_X(0)^\beta h(\frac12)$, and $h:Y\to{{\mathbb R}}^+$ is the density for $\mu$.

See [@Gouezel-stable Theorem 1.3] for a similar calculation in the case $\alpha<1$. Recall (see for example [@MT12 Proposition 11.12]) that the first return time $\tau:Y\to{{\mathbb Z}}^+$ satisfies $$\begin{aligned}
 \label{eq-PMtau}
\mu(\tau>n)= c_1 n^{-\beta}(1+O(n^{-\beta})),\end{aligned}$$ where $c_1=\frac14\beta^\beta h(\frac12)$.

If $\tau(y)=k$, then write $$\varphi(y)=\sum_{j=0}^{k-1} \varphi_X(f^jy)=\varphi_X(y)+(k-1)\varphi_X(0)
+\sum_{j=1}^{k-1}(\varphi_X(f^jy)-\varphi_X(0)),$$ where $f^jy=O((k-j)^{-\beta})$ for $j=1,\dots,k-1$. It follows that $$\Bigl|\sum_{j=1}^{k-1}(\varphi_X(f^jy)-\varphi_X(0))\Bigr|\le
|\varphi_X|_\eta\sum_{j=1}^{k-1}|f^jy)|^\eta \ll k^{1-\eta\beta}.$$ Hence $\varphi(y)=k(\varphi_X(0)+O(k^{-\beta\eta}))=
\varphi_X(0)\tau(y)(1+O(\tau(y)^{-\beta\eta})$. This combined with  yields the result.

\[cor-PM\] Suppose that $X=[0,1]$ and that 