Re: M-set: from internal angles to external

Roger Bagula napisał(a): 
I have already discussed how to get the angles by solving the Mandelbrot
iterative polynomials. I developed a method based on that.
Here's a program that solves the angles graphically in Mathematica:

Clear[x, y, a, b, f, z]
g[z_] = z^2 + c
nr = Expand[D[NestList[g, c, 6], c]]
c = z
p[z_] = Apply[Times, nr]
z = x + I*y
f[x_, y_] = Re[1/(p[z])]
ContourPlot[ f[x, y], {x, -2.7, 1.5}, {y, -2.1,
    2.1}, PlotPoints -> {300, 300},
    ImageSize -> 600,
     ColorFunction -> (Hue[2#] &)]

What it does essentially is to get the PC(c,n) polynomials
to the 6th iterative level and then, makes a product of them
and takes the inverse of that so that the solutions behave
in a way that shows up as lines in an (implicit) contour plot.
The result is equivalent to the lines called "field"( opposite to equipotential) lines in
"The Beauty of Fractals" page 66, but in greater detail and extended inside the plateau.
Basic to this all is the discussion in a paper by Douady and Hubbard.
My method is much easier than theirs.
What you have to realize is that:
1) the center of the Mandelbrot set is offset from (0,0)
2) The complex plane dynamics behaves in a "fluid mechanics" like way
with potential and field flows
3) That there is a symmetry axis in the x axis
that makes a y to -y symmetry
4) the "***" of the set is the zero angle and the main antenna
( near x=-2) is 1/2 (Pi, 180 degree) so that the two parts of the Farey map above are symmetrical about their angular values.
One way of looking at it is that for z and the conjugate zstar:
z=x+I*y --> angle =Arg[z]
zstar=x-I*y-->angle=Arg[zstar]
Are the same but in opposite directions from the x axis with center offset ( behave like the y sign).

What might help you to understand the dynamics is the Joukowski cylinder flow used in analysis of air flow about wings in aerodynamics.
The Mandelbrot set is topologically a cylinder.
The Joukowski cylinder flow is :
w=a*(z+r^2/z)-Gamma*Log[z]/(2*Pi*I)
and the streamlines behave like the angles.
The set itself is like the potential line ( flow circulation) at some critical value.

adam majewski wrote:


I thought that it would be a simple problem but it seems to be more difficult and I don't know math as much as you.

1.Could you show me some sources about this to let me improve my knowlage ?

2. so if the Linas method is bad , because it gives only approximations of external angles not the exact values ( if I've uderstanded you) do you think that is better method ?

3.could you show how big the the error between approximation and exact value could be ?



Adam Majewski
republika.pl/fraktal

ps:
Linas Vepstas page at Wikipedia is active and contains some intresting informations ( KAM ):
http://en.wikipedia.org/wiki/User:Linas



Hi
thx. Can you show some plots from this program. I don't have Mathemathica ( but I have free Maxima ). Maybe you know how to do it in Maxima ?

Adam Majewski
republika.pl/fraktal
.

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