Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

A parameterization of the period $ 3$ hyperbolic components of the Mandelbrot set


Authors: Dante Giarrusso and Yuval Fisher
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We demonstrate that the period 3 hyperbolic components of the Mandelbrot set consist of the image of the unit disk by the maps

$\displaystyle - \frac{7}{4} - \frac{{20}}{9}{\left[ {\sinh \left( {\omega (z) + \frac{{2k\pi i}}{3}} \right) - \frac{1}{{4\sqrt 5 }}} \right]^2},$

with

$\displaystyle \omega (z) = \frac{1}{3}{\operatorname{Arcsinh}}\left( {\frac{{88 - 27z}}{{80\sqrt 5 }}} \right),$

for $ k = 0,1,2$.

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

  • [1] A. Douady and J. H. Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris 294 (1982). MR 651802 (83m:58046)
  • [2] P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France 47 (1919); 48 (1920).
  • [3] D. Giarrusso, unpublished notes, Cornell University, Ithaca, NY, 1989.
  • [4] E. Netto, Theory of substitutions, Chelsea, New York, 1892.
  • [5] J. Stephenson and D. T. Ridgway, Formulae for cycles in the Mandelbrot set II, Phys. A 190 (1992), 104-116. MR 1191327 (94g:58191)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30D05, 30C10

Retrieve articles in all journals with MSC: 30D05, 30C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1301497-3
Keywords: Iterative dynamics, Mandelbrot set, hyperbolic components, factorization
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society