Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 3 / b / Solution

Assume $J(f)$ is connected. Then the filled Julia set $K(f)$ is connected and full, so its complement $A_{\infty }(f)$ is simply connected. The local Bottcher coordinate therefore extends by the functional equation to a conformal isomorphismIf a finite critical point $c$ lay in $A_{\infty }(f)$, differentiatingat $c$ would giveNeither factor on the right vanishes on the exterior disc under a conformal coordinate, a contradiction. Hence every finite critical point lies in $K(f)$. This is one direction of the connected Julia set criterion for a polynomial.