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Proceedings of the American Mathematical Society
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
MathSciNet review: 1301497
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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$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1301497-3
PII: S 0002-9939(1995)1301497-3
Keywords: Iterative dynamics, Mandelbrot set, hyperbolic components, factorization
Article copyright: © Copyright 1995 American Mathematical Society