make various further assumptions:

-   There is a constant $C_2>0$ such that $|1_a\varphi|_{\theta_0} \le C_2\inf(1_a\varphi)$ for all $a\in\alpha$.

-   There exist two periodic orbits for $f_t$ with periods $\tau_1,\tau_2$ such that $\tau_1/\tau_2$ is Diophantine.

Condition (A1) is automatic for a large class of examples discussed in Subsection \[sec-GM\]. Condition (A2) is sufficient for a rather technical “approximate eigenfunction” criterion of Dolgopyat [@Dolgopyat98b] to be satisfied. This criterion is stated precisely in Definition \[defn-approx\] and holds also for an open dense set of roof functions [@FMT07].

#### Observables

We consider observables $v,w:\tilde Y\to{{\mathbb R}}$ of the following form. Writing $v^u(y)=v(y,u)$, define $|v|_\theta =\sup_{u\in[0,1]}|v^u|_\theta$ and $\|v\|_\theta=|v|_\infty+|v|_\theta$. Then $F_\theta(\tilde Y)$ is the space consisting of those $v\in L^\infty(\tilde Y)$ with $\|v\|_\theta<\infty$.

For $m\ge0$, set $|w|_{\infty,m}=\max_{j=0,\dots,m}|\partial_t^jw|_\infty$. We write $w\in L^{\infty,m}(\tilde Y)$ if $w$ is supported in $Y\times (0,1)$ with $|w|_{\infty,m}<\infty$.

Define $$\rho_{v,w}(t)=\int_{Y^\varphi} v\,w\circ f_t\,d\mu^\varphi,$$ and write $\bar v=\int_{Y^\varphi}v\,d\mu^\varphi$, $\bar w=\int_{Y^\varphi}w\,d\mu^\varphi$. In the finite measure case, the correlation function of $v$ and $w$ is given by $\rho_{v,w}(t)-\bar v\bar w$. We can now state our main theorems.

\[thm-infinite\] Assume that $F:Y\to Y$ is Gibbs-Markov with roof function $\phi:Y\to{{\mathbb R}}^+$ satisfying conditions (A1) and (A2).

-   Suppose that $\varphi$ is nonintegrable and $\mu(\varphi>t)=\ell(t) t^{-\beta}$ where $\beta\in(\frac12,1]$ and 