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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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Abstract | References | Similar Articles | Additional Information

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, Lattès examples, and tuning sequences of matings.

References [Enhancements On Off] (What's this?)

  • [BEK] Xavier Buff, Adam Epstein, and Sarah Koch.
    Twisted matings and equipotential gluings.
  • [Che12] Arnaud Cheritat.
    Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle.
    preprint, arXiv:1202.4188v1, 2012.
  • [DH82] Adrien Douady and John Hamal Hubbard.
    Itération des polynômes quadratiques complexes.
    C. R. Acad. Sci. Paris Sér. I Math., 294(3):123-126, 1982. MR 651802 (83m:58046)
  • [DH93] Adrien Douady and John H. Hubbard.
    A proof of Thurston's topological characterization of rational functions.
    Acta Math., 171(2):263-297, 1993. MR 1251582 (94j:58143)
  • [Dou83] Adrien Douady.
    Systèmes dynamiques holomorphes.
    In Bourbaki seminar, Vol. 1982/83, volume 105 of Astérisque, pages 39-63. Soc. Math. France, Paris, 1983. MR 728980 (85h:58090)
  • [Dyn] Cornell Dynamics
  • [Eps] Adam Epstein.
    Quadratic mating discontinuity,
    in preparation.
  • [HL04] Peter Haïssinsky and Tan Lei.
    Convergence of pinching deformations and matings of geometrically finite polynomials.
    Fund. Math., 181(2):143-188, 2004. MR 2070668 (2005e:37106)
  • [HS94] John H. Hubbard and Dierk Schleicher.
    The spider algorithm.
    In Complex dynamical systems (Cincinnati, OH, 1994), volume 49 of Proc. Sympos. Appl. Math., pages 155-180. Amer. Math. Soc., Providence, RI, 1994. MR 1315537
  • [Kaw] Tomoki Kawahira.
    Otis fractal program: [˜kawahira/
  • [Lei92] Tan Lei.
    Matings of quadratic polynomials.
    Ergodic Theory Dynam. Systems, 12(3):589-620, 1992. MR 1182664 (93h:58129)
  • [Luo95] Jiaqi Luo.
    Combinatorics and holomorphic dynamics: captures, matings, Newton's method.
    Ph.D. thesis, Cornell University, 1995.
  • [Mil99] John Milnor.
    Dynamics in one complex variable.
    Friedr. Vieweg & Sohn, Braunschweig, 1999.
    Introductory lectures. MR 1721240 (2002i:37057)
  • [Mil04] John Milnor.
    Pasting together Julia sets: a worked out example of mating.
    Experiment. Math., 13(1):55-92, 2004. MR 2065568 (2005c:37087)
  • [Ree92] Mary Rees.
    A partial description of parameter space of rational maps of degree two. I.
    Acta Math., 168(1-2):11-87, 1992. MR 1149864 (93f:58205)
  • [Sel10] Nikita Selinger.
    Thurston's pullback map on the augmented Teichmuller space and applications.
    preprint, arXiv:1010.1690v1, 2010.
  • [Shi00] Mitsuhiro Shishikura.
    On a theorem of M. Rees for matings of polynomials.
    In The Mandelbrot set, theme and variations, volume 274 of London Math. Soc. Lecture Note Ser., pages 289-305. Cambridge Univ. Press, Cambridge, 2000. MR 1765095 (2002d:37072)
  • [Wit88] B. Wittner.
    On the Bifurcation Loci of Rational Maps of Degree Two.
    Ph.D. thesis, Cornell University, 1988. MR 2636558
  • [YZ01] Michael Yampolsky and Saeed Zakeri.
    Mating Siegel quadratic polynomials.
    J. Amer. Math. Soc., 14(1):25-78 (electronic), 2001. MR 1800348 (2001k:37064)

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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.

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