Let $X$ be a compact [metric space](/page/Metric%20Space), let $T:X\to X$ be a topologically mixing uniformly expanding Markov map, and let $m$ be the reference conformal measure used to define the Ruelle transfer operator. Suppose this operator has a strictly positive invariant density $h$ with $\int_X h\,dm=1$, and set $d\mu=h\,dm$, the corresponding Gibbs or absolutely continuous invariant probability measure. Assume there is a Banach algebra $\mathcal B_0\subset L^\infty(\mu)$ of regular observables such that $h\in\mathcal B_0$, constants belong to $\mathcal B_0$, the inclusion $\mathcal B_0\hookrightarrow L^2(\mu)$ is continuous, and the normalized transfer operator

\begin{align*} \mathcal L_\mu g=h^{-1}\mathcal L(hg) \end{align*}

has a simple leading eigenvalue $1$ and a spectral gap on $\mathcal B_0$. Let $f\in\mathcal B_0$ be real-valued with $\int_X f\,d\mu=0$, and assume that the twisted operators $\mathcal L_{\mu,t}g=\mathcal L_\mu(e^{itf}g)$ act boundedly on the complexification of $\mathcal B_0$ and form an analytic perturbation of $\mathcal L_\mu$ for $t$ near $0$. Then there exists $\sigma^2\ge 0$ such that

\begin{align*} \frac{S_nf}{\sqrt n}\xrightarrow{d}\mathcal N(0,\sigma^2). \end{align*}

Moreover,

\begin{align*} \sigma^2=\int_X f^2\,d\mu+2\sum_{n=1}^{\infty}\int_X f(f\circ T^n)\,d\mu, \end{align*}

and $\sigma^2=0$ precisely when there exists $u\in\mathcal B_0$ such that $f=u-u\circ T$ $\mu$-a.e.