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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