Location of the atoms of the Mandelbrot set

[Henning Kühn]

I am looking for an explanation for the location of the atoms which are
attached to the boundary of the cardioid of the Mandelbrot set.

For any rational numbers phi=n/m (with n,m natural numbers) an atom with an
attractive orbit of the period m is attached to the main cardioid at the
point c=exp(i*phi)/2-exp(i*2*phi)/4. But how can we derive this fact?

The boundary of the main (period one) cardioid can be derived by using the
condition for indifferent fixed points:

Abs(d(f(z))/dz)=Abs(2z)=1 (with f(z)=z^2+c)

So we can write

z=exp(i*phi)/2

together with the fixed point equation

z=z^2+c

we get

c=exp(i*phi)/2-exp(i*2*phi)/4

This curve drawn in the complex plane results in the heart-like shape of the
cardioid as you can see for example here:

http://en.wikipedia.org/wiki/Mandelbrot_set

The boundary of the period two bud

Abs(1+c)=1/4

corresponding to a circle with radius 1/4 centered at c=-1 can be derived in
a similar way.

But how can we derive the general case that an atom with an attractive orbit
of the period m is attached to the main cardioid at the point
c=exp(i*n/m)/2-exp(i*2*n/m)/4?

Many thanks for answers.

Henning

[Henning Kühn]

Post by Henning Kühn
I am looking for an explanation for the location of the atoms which are
attached to the boundary of the cardioid of the Mandelbrot set.
For any rational numbers phi=n/m (with n,m natural numbers) an atom with
an attractive orbit of the period m is attached to the main cardioid at
the point c=exp(i*phi)/2-exp(i*2*phi)/4. But how can we derive this fact?

I mean phi=2*Pi*n/m of course.