with indifferent fixed points, in particular the Pomeau-Manneville intermittency maps [@PomeauManneville80] which are uniformly expanding away from an indifferent fixed point at $0$. To fix notation, we focus on the version studied by Liverani [*et al.*]{} [@LiveraniSaussolVaienti99]: $$\begin{aligned}
 \label{eq-LSV}
fx=\begin{cases} x(1+2^\alpha x^\alpha), & 0<x<\frac12 \\ 2x-1 & \frac12<x<1
\end{cases}.\end{aligned}$$

When $\alpha=0$, this is the doubling map, and Lebesgue measure is invariant, ergodic and exponentially mixing. For $\alpha\in(0,1)$, there is still a unique ergodic invariant probability measure $\mu$ absolutely continuous with respect to Lebesgue measure, but the rate of mixing is polynomial: $\int_X v\,w\circ f^n \,d\mu-\int_X v\,d\mu \int_Xw\,d\mu \le 
C_{v,w}n^{-(\beta-1)}$ where $\beta=\frac{1}{\alpha}$ for all $w\in L^\infty(X)$ and all $v$ sufficiently regular (for example, Hölder continuous). Hu [@Hu04] proved that this decay rate is optimal; a special case of the theory of [@Gouezel04; @Sarig02].

For $\alpha\ge1$, we are in the situation of infinite ergodic theory. There no longer exists an absolutely continuous invariant probability measure, but there is a unique (up to scaling) $\sigma$-finite, absolutely continuous invariant measure $\mu$. Previous studies established [*pointwise dual ergodicity*]{}: $a_n^{-1}\sum_{j=1}^nL^jv \to {\rm const}\,\int_X v\,d\mu$ almost everywhere for all $v\in L^1(X)$, where $a_n=n^{\beta}$ for $\beta\in(0,1)$ and $a_n=n/\log n$ when $\beta=1$.

An important refinement is the limit theorem 