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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A parameterization of the period $3$ hyperbolic components of the Mandelbrot set
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by Dante Giarrusso and Yuval Fisher PDF
Proc. Amer. Math. Soc. 123 (1995), 3731-3737 Request permission

Abstract:

We demonstrate that the period 3 hyperbolic components of the Mandelbrot set consist of the image of the unit disk by the maps \[ - \frac {7}{4} - \frac {{20}}{9}{\left [ {\sinh \left ( {\omega (z) + \frac {{2k\pi i}}{3}} \right ) - \frac {1}{{4\sqrt 5 }}} \right ]^2},\] with \[ \omega (z) = \frac {1}{3}{\operatorname {Arcsinh}}\left ( {\frac {{88 - 27z}}{{80\sqrt 5 }}} \right ),\] for $k = 0,1,2$.
References
  • 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). MR 651802
  • P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France 47 (1919); 48 (1920). D. Giarrusso, unpublished notes, Cornell University, Ithaca, NY, 1989. E. Netto, Theory of substitutions, Chelsea, New York, 1892.
  • John Stephenson and Douglas T. Ridgway, Formulae for cycles in the Mandelbrot set. II, Phys. A 190 (1992), no. 1-2, 104–116. MR 1191327, DOI 10.1016/0378-4371(92)90080-A
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 3731-3737
  • MSC: Primary 30D05; Secondary 30C10
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1301497-3
  • MathSciNet review: 1301497