by [@LiveraniSaussolVaienti99], and their suspensions in the continuous time case. Specifically, define the interval maps $f:X\to X$, $X=[0,1]$, $$\begin{aligned}
 \label{eq-LSV}
f(x)=\begin{cases} x(1+2^\alpha x^\alpha), & 0<x<\frac12 \\ 2x-1 & \frac12<x<1
\end{cases},\end{aligned}$$ where $\alpha>0$. There is a unique (up to scaling) $\sigma$-finite absolutely continuous invariant measure and this measure is finite if and only if $\alpha<1$. Such maps have a uniformly expanding (or Gibbs-Markov, see Section \[sec-main\] for precise definitions) first return map to the set $Y=[\frac12,1]$. Set $\beta=1/\alpha$ and $$\begin{aligned}
 \label{eq-xixi}
\xi_\beta(t)  =\begin{cases}
t^{-\beta} & \beta>2 \\ 
(\log t)t^{-2} & \beta=2 \\ 
t^{-(2\beta-2)} & 1<\beta<2 \end{cases}, \quad
\xi_{\beta,\epsilon}(t) = 
\begin{cases}  t^{-(\beta-\epsilon)}, &  \beta\ge 2 \\
t^{-(2\beta-2)}, & 1<\beta<2 \end{cases}.\end{aligned}$$

#### Discrete time, finite measure

For $\alpha\in(0,1)$, it follows from [@Young99; @Hu04] that correlations decay like $n^{-(\beta-1)}$: $$\int_X v\,w\circ f^n-\int_X v\int_X w
=O(n^{-(\beta-1)}),$$ for all $v:X\to{{\mathbb R}}$ Hölder and $w:X\to{{\mathbb R}}$ lying in $L^\infty$. This decay rate is sharp [@Gouezel-sharp; @Sarig02]: $$\int v\,w\circ f^n-\int_X v\int_X w
=c\int_X v\int_X w\; n^{-(\beta-1)}
+O(\xi_{\beta}(n)),$$ for all observables $v,w$ supported in a compact subset of $(0,1]$ with $v$ Hölder and $w$ in $L^\infty$. Here $c$ is a positive constant depending only on $\alpha$.

#### Discrete time, infinite measure

For $\alpha\in(1,2)$, we showed [@MT12] that there is 