Boettcher map for complement of M-set




Boettcher coordinate ( complex number) = complex potential
It creates polar coordinate system of complement of M-set.

Log|Phi| made with program by Wolf Jung

Arg(Phi) made with program by Wolf Jung


Image made with ultrafractal and MMF3 colouring 3.8 (Colourings for Ultrafractal 3+)formula (Field line) by Dave Makin (Makin' Magic)

compare it with: Inigo Quilez images ,  especially Phase of phi(c)
or The Mandelbrot Function 2   and mb1 by John J. G. Savard


Jungreis algorithm gives inverse of Boettcher function.
Point at infinity is fixed point in the parameter plane
It is a superatractive fixed point of mandelbrot function
Complement of M-set is a set of points c in the parameter plane for which iteration of Z0=0 tends to infinity in the dynamical plane.

Phi M  = Boettcher map :


c --> Phi M(c)
C-M --> C-D
it is a conformal  mapping  ( isomorphism )

PhiM(c) = lim (Zn ^ (0.5^n) )
PhiM(c) = lim (FM (c,c,n) ^ (0.5^n) )

phiM(c,n) =
= c * ( 1 + c/c2)1/2* (1 + c/(Fm(c,c))2)1/22*...*(1 + c / (Fn-1(c,c))2)1/2n
= c * Product (( 1 + c/ F M(c,c,i)2)1/2i)  for i = 1 to n
= c * Product (( 1 + c/ Zi2)1/2i)  for i = 1 to n

where F M(c,c,i) it is a Mandelbrot map

It goes like this :

phiM(c,0)= c
phiM(c,1)= c * ( 1 + c/c2)1/2
phiM(c,2) = c * ( 1 + c/c2)1/2 *  (1 + c/(F(c,c))2)1/22

Field lines of complex potential = External ray of angle  t :
R(t) = { Phi M-1 ( r* e 2*Pi* t*i) : 1 <r < infinity }

level curve of potential  r    = equipotential:
= { Phi M-1 ( r* e 2*Pi* t*i) : 0 <=t <= 1}



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Autor: Adam Majewski
adammaj1-at-o2-dot-pl
Feel free to e-mail me. (:-))

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