Conjugacy between polynomial basins

In this article, we study the properties of conjugacies between polynomial basins. For any conjugacy, there is a quasiconformal conjugacy in the same homotopy class minimizing the dilatation. We compute the precise value of the minimal dilatation. The quasiconformal conjugacy minimizing the dilatation is not unique in general. We give a necessary and sufficient condition when the extremal map is unique.