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

TL;DR

This paper analyzes the geometric shapes of elementary domains within the Mandelbrot Set, revealing they are nearly ideal circles and cardioids, and explains the differences between root and descendant domains through explicit calculations.

Contribution

It provides explicit calculations of elementary domain shapes in the Mandelbrot Set and clarifies the geometric differences between root and descendant domains.

Findings

Elementary domains are nearly ideal circles and cardioids.

Descendant domains have one less cusp than root domains.

Phase transitions and overlaps between domains are explicitly demonstrated.

Abstract

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.