I was tring to do this exercise from Branner and Fagella book on Quasiconformal Surgery.
Suppose $P$ is a polynomial with a superattracting fixed point, say $\alpha$, whose immediate basin, $\mathcal{A}^{\circ}(α)$, contains no other critical point than itself. Let $U$ be a Fatou component which is mapped to $\mathcal{A}^{\circ}(α)$ with degree $k>2$ (i.e. containing at least two critical points).
Prove that there exists a polynomial $Q$ and a quasiconformal homeomorphism $\phi : \mathbb{C} \rightarrow \mathbb{C}$ conjugating $P$ and $Q$ on a neighbourhood of the respective Julia sets, and in all Fatou components except U, such that $\phi(U)$ contains a unique critical point of multiplicity $k-1$ which is mapped to $\phi(\alpha)$.
I was able to construct something that conjugated $P$ and $Q$ outside $U$ and the basin of attraction $\mathcal{A}^{\circ}(α)$. I do not get how I can do something only on U knowing that the applications send it in $\mathcal{A}^{\circ}(α)$.