the first return map is Gibbs-Markov. The latter includes parabolic rational maps of the complex plane [@ADU93].

#### Notation

We use “big O” and $\ll$ notation interchangeably, writing $A_n=O(a_n)$ or $A_n\ll a_n$ as $n\to\infty$ if there is a constant $C>0$ such that $\|A_n\|\le Ca_n$ for all $n\ge1$ (for $A_n$ operators and $a_n\ge0$ scalars). We write $A_n\sim c_nA$ as $n\to\infty$ if $\lim_{n\to\infty}\|A_n/c_n-A\|=0$ (for $A_n,A$ operators and $c_n>0$ scalars).

General framework {#sec-ORT}
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Let $(X,\mu)$ be an infinite measure space, and $f:X\to X$ a conservative measure preserving map. Fix $Y\subset X$ with $\mu(Y)\in(0,\infty)$ and rescale $\mu$ so that $\mu(Y)=1$. Let $\varphi:Y\to{{\mathbb Z}}^+$ be the first return time $\varphi(y)=\inf\{n\ge1:f^ny\in Y\}$ and define the first return map $F=f^\varphi:Y\to Y$.

The return time function $\varphi:Y\to{{\mathbb Z}}^+$ satisfies $\int_Y \varphi\,d\mu=\infty$. Our crucial assumption is that the tail probabilities are regularly varying:

-   $\mu(y\in Y:\varphi(y)>n)=\ell(n)n^{-\beta}$ where $\ell$ is slowly varying and $\beta\in(0,1]$.

Recall that the transfer operator $R:L^1(Y)\to L^1(Y)$ for the first return map $F:Y\to Y$ is defined via the formula $\int_Y Rv\,w\,d\mu = \int_Y v\,w\circ F\,d\mu$, $w\in L^\infty(Y)$. Let ${{\mathbb D}}=\{z\in{{\mathbb C}}:|z|<1\}$ and $\bar{{\mathbb D}}=\{z\in{{\mathbb C}}:|z|\le1\}$. Given $z\in\bar{{\mathbb D}}$, we define $R(z):L^1(Y)\to L^1(Y)$ to be the operator $R(z)v=R(z^\varphi v)$. Also, for each $n\ge1$, we define $R_n:L^1(Y)\to 