to describe our main results in the setting of AFN maps $f:X\to X$, though our general theory goes much further, as described later on. Let $X'\subset X$ denote the complement of the indifferent fixed points. For any compact subset $A\subset X'$, the construction in [@Zweimuller98] yields a suitable first return set $Y$ containing $A$. Fix such a set $Y$ with first return time function $\varphi:Y\to{{\mathbb Z}}^+$. Then we assume that the tail probabilities are regularly varying: $\mu(\varphi>n)=\ell(n)n^{-\beta}$ where $\beta\in(0,1]$ and $\ell(x)$ is slowly varying ($\ell:(0,\infty)\to(0,\infty)$ is measurable and $\lim_{x\to\infty}\ell(\lambda x)/\ell(x)=1$ for all $\lambda>0$). (For , $\ell$ is asymptotically constant and $\beta=\frac{1}{\alpha}$.)

Now we can state our results for AFN maps. Set $d_\beta ={{\textstyle}\frac{1}{\pi}}\sin\beta\pi$ for $\beta\in(0,1)$ and $d_1=1$. Define $m(x)=\ell(x)$ for $\beta\in(0,1)$ and $m(x)= \sum_{j=1}^{[x]}\ell(j)j^{-1}$ for $\beta=1$.

\[thm-AFN\] Suppose that $f:X\to X$ is a topologically mixing AFN map with $\sigma$-finite absolutely continuous invariant measure $\mu$ and regularly varying tail probabilities. Consider observables of the form $v=\xi u$ where $\xi$ is $\mu$-integrable and of bounded variation on $X$, and $u$ is Riemann integrable.

-   If $\beta\in(\frac12,1]$, then $\lim_{n\to\infty} m(n)n^{1-\beta}L^n v=d_\beta\int_X v\,d\mu$ uniformly on compact subsets of $X'$.

-   If $\beta\in(0,\frac12]$, then there is a subset $E\subset{{\mathbb N}}$ of zero density such that 