Planes of M and J or F sets
Planes for Mandelbrot function F_{c}(z)=z*z+c (c is complex parameter, z is complex variable)
 dynamic planes = planes for Julia or Fatou sets ( c is constant; z_{0}=z ; z is a variable, )
 parameter planes = planes for Mandelbrot Set ( z_{0}:=0 ; c is variable )
 cplane ( plain parameter plane )
 1/c plane ( inverted parameter plane)
 1/(c0.25) plane ;
 1/(c+1.40115) plane
 1/(c+2.0) plane
 unrolled ( flatten) plane
map c = r/2 e^{ip} (1  r/2 e^{ip}) with r running from 1 at the bottom to 5 at the top,
and p running from 1.1 pi to +1.1 pi left to right. The cardioid is then the straight line r=1 ( unrolled or flatten the cardioid)
 Mercator plane by David Madore :
"... it is a log map toward the target point (or, as some might say, a Mercator projection with the target point as South pole
and complex infinity as North pole); horizontally it is periodic and I have placed two periods side to side,
whereas vertically it extends to infinity at the top and at the bottom,
which corresponds to zooming infinitely far out or in, at a factor of exp(2*Pi) = 535.5
for every size of a horizontal period. Horizontal lines ( parallels ) on the log map correspond to concentric circles
around the target point, and vertical lines to radii emanating from it; and the anamorphosis preserves angles."
"Images of the Mandelbrot set with a logarithmic projection around a point c0: z> (log(zc0), arg(zc0)).
The idea for this kind of logarithmic map of the set is from David Madore.
While it is a bit of a stretch to call it a Mercator projection, it sounds better than just a log scale map.
If the Mandelbrot set is assumed to be about metersized these zooms reach the size of an atomic nucleus." ( Arenamontanus / Anders Sandberg )

multiplier plane:
multiplier map of hyperbolic component converts component into unit circle
Multiplier planes for main cardioid ( H_{0})
 m = M(c) = 1  sqrt(1  4*c)
It is a correspondence between the interior of the main cardioid and the interior of the unit disc D( disk of radius 1 centered at the origin)
 "square" parametrisation:
"c = 1/4  a2
Z_{n+1} = Z_{n}^{2} + 1/4  a^{2}
the main cardioid of the Mset turns into a circle with radius r = 1/2.
A primary bulb attaches to the main circle at an internal angle:
phi = 2 *p *m/n
where m/n is rotation number (e.g. 1/2 > 180^{o}, 1/3 > 120^{o} and 1/4 > 90^{o}) "
( quotation from: Rotation Numbers and Internal angles of the Mandelbrot bulbs by Robert L. Devaney
Planes for F_{lambda}(z) = lambda* z*(1z):
 lambda
 1/lambda
 The 1/(lambda1)
Planes for F_{m}(z) =m*z+(1m)*z*z
 m plane
here the (finite) fixed points of F_{m}(z) are:
Z1=0
Z2=1
Bibliography:
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Autor: Adam Majewski adammaj1ato2dotpl
Feel free to email me. (:))
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