×

The Medusa algorithm for polynomial matings. (English) Zbl 1291.37054

Summary: 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.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37M99 Approximation methods and numerical treatment of dynamical systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Xavier Buff, Adam Epstein, and Sarah Koch. Twisted matings and equipotential gluings. submitted. · Zbl 1408.37084
[2] Arnaud Cheritat. Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle. preprint, arXiv:1202.4188v1, 2012. · Zbl 1320.37023
[3] Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123 – 126 (French, with English summary). · Zbl 0483.30014
[4] Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263 – 297. · Zbl 0806.30027 · doi:10.1007/BF02392534
[5] Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39 – 63 (French). · Zbl 0532.30019
[6] Cornell Dynamics [http://www.math.cornell.edu/\~dynamics].
[7] Adam Epstein. Quadratic mating discontinuity, in preparation.
[8] Peter Haïssinsky and Lei Tan, Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math. 181 (2004), no. 2, 143 – 188. · Zbl 1048.37045 · doi:10.4064/fm181-2-4
[9] John H. Hubbard and Dierk Schleicher, The spider algorithm, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 155 – 180. · Zbl 0853.58088 · doi:10.1090/psapm/049/1315537
[10] Tomoki Kawahira. Otis fractal program: [http://www.math.nagoya-u.ac.jp/\~kawahira/programs/otis.html].
[11] Lei Tan, Matings of quadratic polynomials, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 589 – 620. · Zbl 0756.58024 · doi:10.1017/S0143385700006957
[12] Jiaqi Luo. Combinatorics and holomorphic dynamics: captures, matings, Newton’s method. Ph.D. thesis, Cornell University, 1995.
[13] John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. · Zbl 0946.30013
[14] John Milnor, Pasting together Julia sets: a worked out example of mating, Experiment. Math. 13 (2004), no. 1, 55 – 92. · Zbl 1115.37051
[15] Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math. 168 (1992), no. 1-2, 11 – 87. · Zbl 0774.58035 · doi:10.1007/BF02392976
[16] Nikita Selinger. Thurston’s pullback map on the augmented Teichmuller space and applications. preprint, arXiv:1010.1690v1, 2010. · Zbl 1298.37033
[17] Mitsuhiro Shishikura, On a theorem of M. Rees for matings of polynomials, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 289 – 305. · Zbl 1062.37039
[18] Ben Scott Wittner, On the bifurcation loci of rational maps of degree two, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.) – Cornell University.
[19] Michael Yampolsky and Saeed Zakeri, Mating Siegel quadratic polynomials, J. Amer. Math. Soc. 14 (2001), no. 1, 25 – 78 (electronic). · Zbl 1050.37022 · doi:10.1090/S0894-0347-00-00348-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.