In the dynamics of one complex variable, it is well-known that some neighborhood of any attracting (but not super-attracting) fixed point is linearized under some holomorphic change of coordinates. However, this linearization cannot be extended to the `hole' immediate basin of attraction, due to the existence of critical points in the basin.
In this talk, we consider an arbitrary rational map $f$ on the Riemann sphere, an attracting fixed point $p$ of $f$, and the connected component $U$ of the Fatou set $F(f)$ containing $p$, and aim to construct all the holomorphic equivalence classes of $(f, U)$ for any given covering degree of $f: U \to U$ and any given number of critical points on $U$, regardless of whether $U$ is simply connected or infinitely connected.
If time permits, we also consider the parabolic case and the super-attracting case.