obtain higher order expansions of $L^nv$ on compact sets for bounded variation observables $v$ supported on a compact subset of $(0,1]$. For example, in the specific case of , $\beta\in(\frac12,1)$, we prove that $|n^{1-\beta}\int_0^1 v\,w\circ f^n\,d\mu-\int_0^1 v\,d\mu\int_0^1 w\,d\mu|
\le Cn^{-\gamma}$ where $\gamma=\min\{1-\beta,\beta-\frac12\}$. The rate of mixing is sharp for $\beta\ge\frac34$, and we obtain precise second order asymptotics provided $\beta>\frac34$.

These and related results such as error rates in the Dynkin-Lamperti arcsine laws [@Dynkin61; @Lamperti58] are discussed in Section \[sec-second\].

The remainder of the paper is organised as follows. In Section \[sec-ORT\], we describe the general framework for our results on the renewal operators $T_n$. Sections \[sec-T\], \[sec-Fourier\] and \[sec-half\] contain the proofs for $\frac12<\beta<1$. In Section \[sec-one\], we cover the case $\beta=1$. In Section \[sec-pde\], we give a particularly elementary, self-contained, proof of pointwise dual ergodicity for all $\beta$. In Section \[sec-zerohalf\], we prove our results for $0<\beta\le\frac12$. In Section \[sec-second\], we formulate and prove results on higher order asymptotics. In Section \[sec-L\], we show how to pass from $T_nv=1_YL^n(1_Yv)$ to $L^n$. Finally in Section \[sec-examples\], we show that our theory applies to large classes of examples including AFN maps (in particular, we prove Theorem \[thm-AFN\]) and systems for which 