35,334 research outputs found
On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?
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 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.Comment: 65 pages, 30 figure
Carrots for dessert
Carrots for dessert is the title of a section of the paper `On
polynomial-like mappings' by Douady and Hubbard. In that section the authors
define a notion of dyadic carrot fields of the Mandelbrot set M and more
generally for Mandelbrot like families. They remark that such carrots are small
when the dyadic denominator is large, but they do not even try to prove a
precise such statement. In this paper we formulate and prove a precise
statement of asymptotic shrinking of dyadic Carrot-fields around M. The same
proof carries readily over to show that the dyadic decorations of copies M' of
the Mandelbrot set M inside M and inside the parabolic Mandelbrot set shrink to
points when the denominator diverge to infinity.Comment: 21 pages, 2 figure
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