Conjugacy between polynomial basins

TL;DR

This paper investigates the properties of conjugacies between polynomial basins, focusing on quasiconformal conjugacies that minimize dilatation, and provides conditions for their uniqueness.

Contribution

It computes the exact minimal dilatation for conjugacies and characterizes when the extremal quasiconformal map is unique.

Findings

Computed the precise minimal dilatation value.

Established necessary and sufficient conditions for extremal map uniqueness.

Showed that the minimal dilatation conjugacy is not always unique.

Abstract

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.