On the parabolic Fatou domains

Let $f$ be a rational map with an infinitely-connected fixed parabolic Fatou domain $U$. We prove that there exists a rational map $g$ with a completely invariant parabolic Fatou domain $V$, such that $(f,U)$ and $(g,V)$ are conformally conjugate, and each non-singleton Julia component of $g$ is a Jordan curve which bounds a superattracting Fatou domain of $g$ containing at most one postcritical point. Furthermore, we show that if the Julia set of $f$ is a Cantor set, then the parabolic Fatou domain can be perturbed into an attracting one without affecting the topology of the Julia set.