The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Computing the Scaling Factor

If the convergence is sufficiently uniform (so the derivatives converge), we have

(d/de)f_n(-2 + e/r_n)|_e=0 = g_n'(e)|_e=0 = g'(e)|_e=0

By the chain rule, the equation becomes

((d/dc)f_n(c)|_c=-2)*((d/de)(-2 + e/r_n)|_e=0) = g'(0).

Motivated by the slope of f₃ at the 3-cycle midget cardioid center, we take

g'(0) = -1.

With this, we find

r_n = -f_n'(-2)

The relation

f_n+1(c) = (f_n(c))² + c

implies

f_n+1'(c) = 2f_n(c)f_n'(c) + 1.

Using f₁'(c) = 1, f₁(-2) = -2, and f_n(-2) = 2 for N > 1, it is easy to see

f_n'(-2) = -4ⁿ/6 - 1/3,

for n > 1. Dropping the -1/3, small compared to the other term for large n, we take the scaling factor to be

r_n = 4ⁿ/6.

Return to Hurwitz-Robucci scaling.