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Abstract: Expressions are given for the number of hyperbolic components, cardioids, and discs which are associated with N-cycles for the Mandelbrot set. We also discuss the distribution of hyperbolic components along the real axis.
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Young Hee Geum, Young Ik Kim : An improved computation of component centers in the degree-n bifurcation set. The Korean Journal of Computational & Applied Mathematics Volume 10 , Issue 1-2 (September 2002) Pages: 63 - 73 2002
Young Hee Geum, Young Ik Kim : A STUDY ON COMPUTATION OF COMPONENT CENTERS IN THE DEGREE- n BIFURCATION SET. International Journal of Computer Mathematics, Volume 80, Issue 2 February 2003 , pages 223 - 232
Abstract
The governing equation locating component centers in the degree- n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor (n - 1) . Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2\leq n \leq 25 and show a remarkably improved computation. Our study extends the results given by Peitgen and Richter [11].
Keywords: Bifurcation; Component Centers; Degree- n Bifurcation Set; Mandelbrot Set; Newton's Method

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Author: Adam Majewski adammaj1-at-o2-dot-pl

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2005-07-02

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