Unmating of rational maps, sufficient criteria and examples

TL;DR

This paper investigates when a postcritically finite rational map can be decomposed into polynomials via mating, providing a sufficient condition and an explicit algorithm for unmatting, with several examples.

Contribution

It introduces a sufficient condition for a rational map to be a mating and presents an explicit unmatting algorithm, advancing understanding of polynomial rational map relationships.

Findings

Provided a sufficient condition for rational maps to be matings.

Developed an explicit algorithm to unmate rational maps.

Presented multiple examples demonstrating the unmatting process.

Abstract

Douady and Hubbard introduced the operation of mating of polynomials. This identifies two filled Julia sets and the dynamics on them via external rays. In many cases one obtains a rational map. Here the opposite question is tackled. Namely we ask when a given (postcritically finite) rational map $f$ arises as a mating. A sufficient condition when this is possible is given. If this condition is satisfied, we present a simple explicit algorithm to unmate the rational map. This means we decompose $f$ into polynomials, that when mated yield $f$. Several examples of unmatings are presented.