The Medusa Algorithm for Polynomial Matings

TL;DR

This paper describes the Medusa algorithm for computing the mating of two quadratic polynomials, providing approximations of the resulting rational map and its Julia set, with convergence checked via Thurston's theorem.

Contribution

It details the implementation of the Medusa algorithm for polynomial mating, including output analysis and visualizations, expanding practical tools for complex dynamics.

Findings

Algorithm successfully computes polynomial matings

Generated visualizations of Julia sets and matings

Discussed convergence and specific examples like shared matings

Abstract

The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image of its Julia set. Whether these approximations converge is answered using Thurston's topological characterization of rational maps. This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris, and Kuon Ju Liu. In this paper we describe the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattes examples, and tuning sequences of matings.