The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Locating the Centers of Components

In each component, for one c, called the center of the component, the derivative (F_cⁿ(z_i))' = 0.

By the chain rule

0 = (F_cⁿ)'(z_i) = F_c'(z_n) ... F_c'(z₁)

Because F_c'(z) = 2z, if c is the center of an n-cycle component we have z_i = 0 for some i. Consequently,

0 = F_cⁿ(0) = F_c^n-1(c).

That is, the centers of the n-cycle cmponents are the zeros of the family of polynomials f_n(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.

Return to Hurwitz-Robucci scaling.