Elastic Graphs for Main Molecule Matings

TL;DR

This paper applies Thurston's positive criterion to prove a theorem on the mateability of quadratic polynomials within the main molecule, offering a new approach to classical problems in complex dynamics.

Contribution

It introduces a novel application of Thurston's positive criterion to classical polynomial mating problems, providing a new proof and potential insights into higher degree cases.

Findings

New proof of quadratic polynomial mateability within the main molecule

Application of Thurston's positive criterion to classical problems

Potential framework for understanding higher degree polynomial mateability

Abstract

Recent work of Dylan Thurston gives a condition for when a post-critically finite branched self-cover of the sphere is equivalent to a rational map. We apply D. Thurston's positive criterion for rationality to give a new proof of a theorem of Rees, Shishikura, and Tan about the mateability of quadratic polynomials when one polynomial is in the main molecule. These methods may be a step in understanding the mateability of higher degree post-critically finite polynomials and demonstrate how to apply the positive criterion to classical problems.