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MathGroup Archive 2004

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Implicit root structures in the Mandelbrot set

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50954] Implicit root structures in the Mandelbrot set
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Wed, 29 Sep 2004 03:15:25 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com


I have used the implicit root structures in the Mandelbrot set
 to gain new visual information about the set.
The Pcn(0) polynomials are in books that talk about quadratic cycles in 
Julias and the Mandelbrot set.
(Complex Dynamics ,Carleson and Gamelin,Springer,1991, page129)
They give a root structure that determines the Mandelbrot set.
I use the differential set as being representative, but also one step simpler
and having fewer roots overall which becomes important as
the root numbers go up as powers of two. 
I'm seeing the structure in the antenna region
where it is supposed to be.
I'm seeing structures inside the plateau that are definitely there.
The important thing about this method is that it isn't iteration dependent.
These are the roots that show up at any iteration level.
The fractal roughness/ complexity seems to set in after n=6
and just gets really large past n=7! My first try at n=7 and I ran out 
of kernel memory. I haven't tried n=8.
There appears to be a "bulb" inside the cardioid that makes it more like 
a Limacon of Pascal.
Other methods give evidence of this kind of structure as well, but the iteration 
dependence is hard to overcome.
I suppose I haven't really overcome it, just stepped outside of it...

Mathematica program:
Clear[x,y,a,b,f,z]
g[z_]=z^2+c;
nl=NestList[g,c,7];
nr=D[nl,c];
c=z;
p[z_]=Apply[Times,nr];
z=x+I*y;
p[z_]=Apply[Times,nr];
f[x_,y_]=Re[1/(p[z])];
ImplicitPlot[f[x,y]==0, {x, -2.5, 1}, {y, -1.75,1.75},
  PlotPoints -> {100, 100}]
ImplicitPlot[f[x,y]==0, {x, -2.5, -1.}, {y, -0.75,0.75},
  PlotPoints -> {100, 100}]
Plot3D[f[x,y], {x, -2.5, 1}, {y, -1.75,1.75}, PlotPoints -> {145, 145},
  Mesh->False,Boxed->False,Axes->False]
Plot3D[f[x,y], {x, -2.5, -1.}, {y, -0.75,0.75}, PlotPoints -> {200, 200},
  Mesh->False,Boxed->False,Axes->False]

Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 



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