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
