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Conformal Geometry and Dynamics

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The Medusa algorithm for polynomial matings

Authors: Suzanne Hruska Boyd and Christian Henriksen
Journal: Conform. Geom. Dyn. 16 (2012), 161-183
MSC (2010): Primary 37F10; Secondary 37M99
Published electronically: June 26, 2012
MathSciNet review: 2943594
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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, Lattès examples, and tuning sequences of matings.

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Additional Information

Suzanne Hruska Boyd
Affiliation: Department of Mathematical Sciences, University of Wisconsin Milwaukee, PO Box 413, Milwaukee, Wisconsin 53201

Christian Henriksen
Affiliation: Department of Mathematics, Building 303, Technical University of Denmark, Denmark – 2800 Kgs. Lyngby, Denmark

Received by editor(s): February 24, 2011
Published electronically: June 26, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.