In each component, for one c, called the center of the component,
the derivative (Fcn(zi))' = 0. |
By the chain rule |
0 = (Fcn)'(zi) =
Fc'(zn) ... Fc'(z1) |
Because Fc'(z) = 2z, if c is the center of an n-cycle component we have
zi = 0 for some i. Consequently, |
0 = Fcn(0) = Fcn-1(c). |
That is, the centers of the n-cycle cmponents are the zeros of the family of polynomials
fn(c) defined by |
f1(c) = c |
fn+1(z) = (fn(c))2 + c |
|
Now we restrict our attention to real values of c, that is, to the part
of the Mandelbrot set along the real axis. |