The Mandelbrot set renormalization

It is evident, that one can apply discussed above "linear" theory to the midgets on complex plane. You see below the period-3 Mandelbrot midget located at c3 = -1.7542 . It is βΛ32 = 52.5334 times smaller then the main M-set. J(0), Rabbit, Cauliflower (and all the rest Julia) midgets shrink Λ3 = -9.29887 times and are "placed " in the usual typical points (c3, r, c) of the M3 midget.

The M4 midget scaling

For the biggest period-4 M-midget Λ4 = -10.55 - 5.448i,   β = 0.7889 - 0.2754i and m = β Λ42 = 96.14 + 68.23i. So this copy is reduced |m| = 117.88 times and rotated by Arg(m) = 35.36o.
Contents   Previous: The Julia set renormalization   Next: Polynomial-like maps
updated 14 Sep 2013