of Thaler [@Thaler95] where the convergence of $a_n^{-1}\sum_{j=1}^nL^jv$ is shown to be uniform on compact subsets of $(0,1]$ for all observables of the form $v=u/h$ where $u$ is Riemann integrable and $h$ is the density.

The results of [@Thaler95] are formulated for Markov maps of the interval with indifferent fixed points. Zweimüller [@Zweimuller98; @Zweimuller00] relaxed the Markov condition and systematically studied a large class of non-Markovian nonuniformly expanding interval maps, called *AFN maps*. (See Section \[sec-AFN\] for a precise definition of AFN map.) In particular, [@Zweimuller98] obtained a spectral decomposition into basic (conservative and ergodic) sets and proved that for each basic set there is a $\sigma$-finite absolutely continuous invariant measure, unique up to scaling. The results in [@Thaler95] on uniform pointwise dual ergodicity were extended in [@Zweimuller00] to the class of AFN maps.

Understanding the asymptotics of $L^n$, rather than $\sum_{j=1}^nL^j$, turns out to be a much more difficult problem, even for . Previously, the sole success in this direction was obtained by Thaler [@Thaler00]. However, the class of systems covered by [@Thaler00] is quite restrictive and includes the family  only for $\beta=1$.

It is this situation that we have sought to redress in this paper. It is convenient 