Asymptotically Gaussian distribution for random perturbations of rotations of the circle

Let $T_{\epsilon,\omega}$ be a self-map of the two dimensional torus $\mathbb T^2$ given by the formula $T_{\epsilon,\omega}\colon(x,y)\to(2x,y+\omega+\epsilon x)\bmod1$. If $\epsilon$ is an irrational number, a version of the functional central limit theorem is formulated for variables of the form $n^{-1/2} \sum_{k=0}^{\infty}f \circ T^k_{\epsilon,\omega}$ where $f$ is a member of a class of real valued functions on $\mathbb T^2$ described in terms of $\epsilon$. The proof will be published elsewhere.