Let $\varphi:Y\to{{\mathbb Z}}^+$ be the first return time $\varphi(y)=\inf\{n\ge1:f^ny\in Y\}$ (finite almost everywhere by conservativity). Let $L:L^1(X)\to L^1(X)$ denote the transfer operator (Perron-Frobenius operator) for $f$ and define $$T_nv=1_YL^n(1_Yv),\enspace n\ge0, \qquad R_nv=1_YL^n(1_{\{\varphi=n\}}v),\enspace n\ge1.$$ Note that $T_n$ and $R_n$ can be viewed as operators on $L^1(Y)$ with $T_0=I$. Thus $T_n$ corresponds to returns to $Y$ and $R_n$ corresponds to first returns to $Y$. The relationship $T_n=\sum_{j=1}^n T_{n-j}R_j$ generalises the classical notion of renewal sequences in probability theory.

In the infinite mean setting of Garsia & Lamperti [@GarsiaLamperti62], a crucial requirement is that the first return probabilities have regularly varying tails. In our setting it is natural to assume that the return time probabilities have regularly varying tails. Indeed, many of the results in the infinite ergodic theory literature rely crucially on such an assumption [@Aaronson81; @Thaler98; @ThalerZweimuller06; @Zweimuller00]. Under this assumption, and certain functional-analytic hypotheses on the operators $R_n$, we obtain detailed results (Theorems \[thm-conv\], \[thm-bound\], \[thm-liminf\]) on the asymptotic behaviour of the operators $T_n$ as $n\to\infty$. This has strong ramifications for the asymptotics of the iterates $L^n$, and hence for the underlying dynamical system.

#### Maps with indifferent fixed points

An important class of examples is provided by interval maps 