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/dε)fn(-2 + ε/rn)|ε=0 = gn'(ε)|ε=0 = g'(ε)|ε=0

By the chain rule, the equation becomes

((d/dc)fn(c)|c=-2)⋅((d/dε)(-2 + ε/rn)|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

rn = -fn'(-2)

The relation

fn+1(c) = (fn(c))2 + c

implies

fn+1'(c) = 2⋅fn(c)⋅fn'(c) + 1.

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

fn'(-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

rn = 4n/6.

Return to Hurwitz-Robucci scaling.