Re: Mandelbrot symmetries--> Electromagnetic analogy

A lot of speculation since renormalization complex dynamics has been
tied to magnetic type crystal geometry:
Does the Mandelbrot set imply some sort of electromagnetic analogy?
When we think of symmetry in physics the first name that jumps up is Weyl and his symmetry groups and his gauge theories.
Suppose we try a gauge type approach ( invariant measure with a gauge function):
f(z,c)=z^2+c
f'(z,c)=f(z,c)+I*g*dh(z)/dz=f(z,c)+I*g*w[z]
such that:
Abs[f(z,c)]^2=Abs[f'(z,c)]^2
I solved it in Mathematica:
w[z_] = D[h[z], z]
f[z_, c_] = Expand[z^2 + c]
f1[z_, c_] = Expand[z^2 + c + I*g*w[z]]
Solve[Abs[f[z, c]]^2 - Abs[f1[z, c]]^2 == 0, w[z]]
w[z]_]=I*(c+z^2+/-Abs[c+z^2])/g
Two solutions exist ( + and -):
h[z_, c_] = Integrate[I*(c+z^2+/-Abs[c+z^2])/g ,{z, 0, zout}]
( this doesn't really integrate in Mathematica! ?)
f'[z,c]=f[z,c]+I*g*D[h[z,c],z]
My best estimate at a linear solution is:
w[z]=a0*z+b0
and gauge constant g=2 times center of gravity = 2*cg

What this basically says is that there exists functions in the variable z such that when added to the Mandelbrot iteration
don't change the outcome; that is they are measure invariant
with the Mandelbrot set. My indication from the Mathematica
notebook work is that such functions "exist" ( two of them).
.

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