# Thurston's algorithm and rational maps from quadratic polynomial matings

## TL;DR

This paper introduces an iterative method using finite subdivision rules and Thurston's algorithm to approximate rational maps arising from quadratic polynomial matings, expanding on previous algorithms with proofs, implementation details, and examples.

## Contribution

It presents a new iterative approach to approximate rational maps from polynomial matings, providing proofs, implementation insights, and practical examples.

## Key findings

- The algorithm successfully approximates rational maps in various cases.
- It extends the Medusa algorithm with improved efficiency.
- The method is most effective under specific settings.

## Abstract

Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the Riemann sphere. Given a pair of polynomials of the form $z^2+c$ that are postcritically finite, there is a fast test on the constant parameters to determine whether this map $F$ exists---but this test is not constructive. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, $F$. This manuscript expands upon results given by the Medusa algorithm in \cite{MEDUSA}. We provide a proof of the algorithm's efficacy, details on its implementation, the settings in which it is most successful, and examples generated with the algorithm.