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)
• 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.
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