L^1(Y)$, $R_nv=R(1_{\{\varphi=n\}}v)$. It is easily verified that $R(z)=\sum_{n=1}^\infty R_nz^n$.

Our assumptions on the first return map $F:Y\to Y$ are functional-analytic in nature. We assume that there is a function space $\mathcal{B}\subset L^\infty(Y)$ containing constant functions, with norm $\|\;\|$ satisfying $|v|_\infty\le \|v\|$ for $v\in\mathcal{B}$, such that for some constant $C>0$:

-   For all $n\ge1$, $R_n:\mathcal{B}\to\mathcal{B}$ is a bounded linear operator with $\|R_n\|\le C\mu(\varphi=n)$.

In particular, $z\mapsto R(z)$ is a continuous family of bounded linear operators on $\mathcal{B}$ for $z\in\bar{{\mathbb D}}$. Since $R(1)=R$ and $\mathcal{B}$ contains constant functions, $1$ is an eigenvalue of $R(1)$. We require:

-   -   The eigenvalue $1$ is simple and isolated in the spectrum of $R(1)$.

    -   For $z\in\bar{{\mathbb D}}\setminus\{1\}$, the spectrum of $R(z)$ does not contain $1$.

Denote the spectral projection corresponding to the simple eigenvalue $1$ for $R(1)$ by $P$. Then $(Pv)(y)\equiv\int_Y v\,d\mu$.

Asymptotics of $T_n$ {#sec-conv}
--------------------

We state our main results for the operators $T_n$. Since $T_nv=1_YL^n(1_Yv)$, we obtain precise results for the convergence of $L^nv$ on $Y$ for observables $v$ supported on $Y$. The restriction to $Y$ is lifted in Section \[sec-L\]. Throughout, we assume regularly varying tails $\mu(\varphi>n)=\ell(n)n^{-\beta}$ and hypotheses (H1) and (H2). Set $d_\beta ={{\textstyle}\frac{1}{\pi}}\sin\beta\pi$ for $\beta\in(0,1)$ and $d_1=1$. Define $m(n)=\ell(n)$ 