Hubbard-Douady potential of Mandelbrot set for Fc(z) = z*z + c






First : thx for Inigo Quilez for help and many informations. (:-))

Idea: Real potential of a point in the complement of Mandelbrot set shows how quickly critical point Z0=0 diverges toward infinity in the dynamical plane .

GM(c) : C - M --> R
where:
M is a Mandelbrot set
GM is Green's function of Mandelbrot set
C is a complex plane
C-M is a complement of M to C
c is a complex number = point of complex plane C
R is a

Definition :
potential of point c = GM(c)
GM(c) = lim ( (1 / 2 n) *   log2 ( Abs(Zn)) )
     where n --> infinity


How to compute potential of Mandelbrot set ?

If c is a member of M
   then GM(c) =0
  else GM(c) = log2( Abs( z) ) + Sum (  (1 / 2 n+1)  * ( log2(Abs(1 + c / ( Fn(z)2) ))) )

gM(c,n) = (log2|Zn|) / (2^n)




Theory:
Take a 2D plane: X x Y := R x R
Take a scalar function of point position
z=F(x,y)
Then one have 2D scalar field.
If field has no rotational component then F(x,y) is potential of that field
And field is irrotational = conservative vector field <==> curl of that vector field is zero
There are an infinite number of potential functions that lead to the same field,
Usually F(x,y) is 0 at infinity but for Mandelbrot set it is 0 in the interior of that set.
GM(M) = 0
Arnaud Chéritat uses negative vallues for potential of M-set in his program DH_Drawer .
EquiPotential lines consists of points which have the same (constant ) potential
Field lines are normal to equipotential lines.
Field lines of potential are external rays

This images are drawn using simplified definition :
Potential ( c, iter) := 0.5 * log2(hypot(Zn))/IntPower(2, iter);
where Zn:= Fc ( Z0) and Z0 := 0


g:=MSetPotential(Zx,Zy,Iteration);
with Bitmap.FirstLine[iY*Bitmap.LineLength+iX] do
begin
B := bG;
G := bG;
R := bG;
//A := 0;
end ;

One can apply Level Set Method to potential images.

theory:
leves set is a set of points of exterior of M-set for which potential is greater then 2k-1
and smaller or equel to 2k
Lk={ c: 2k-1< potential(c) <= 2k}

how to compute it in delphi:
r:= log2(abs(potential));
k:=ceil(r);

The images of Level Set Method for potential ( pLSM )

are similar to LevelSetMethod for escape time ( eLSM).


All above images are made with my program Mandelbrot set explorer.




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

Made with: EditPlus www.editplus.com



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