On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?

Details On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles? On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?

2007-01-25

arxiv.org/abs/hep-th/0701234v1

Int.J.Mod.Phys.A23:3613-3684,2008

Physics

High Energy Physics

High Energy Physics - Theory

65 pages, 30 figures

Scientific paper

10.1142/S0217751X08040330

Direct look at the celebrated "chaotic" Mandelbrot Set in Fig..\ref{Mand2} immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific {\it forest} structure. In /hep-th/9501235 a systematic algebro-geometric approach was developed to the study of generic Mandelbrot sets, but emergency of nearly ideal circles in the special case of the family $x^2+c$ was not fully explained. In the present paper the shape of the elementary constituents of Mandelbrot Set is explicitly {\it calculated}, and difference between the shapes of {\it root} and {\it descendant} domains (cardioids and circles respectively) is explained. Such qualitative difference persists for all other Mandelbrot sets: descendant domains always have one less cusp than the root ones. Details of the phase transition between different Mandelbrot sets are explicitly demonstrated, including overlaps between elementary domains and dynamics of attraction/repulsion regions. Explicit examples of 3-dimensional sections of Universal Mandelbrot Set are given. Also a systematic small-size approximation is developed for evaluation of various Feigenbaum indices.

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