The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Renormalized Polynomials

On the left we have the graphs of f1, f2, and f3.

The zero of f1 is the center of the big cardioid, at c = 0. Note this is also a zero of f2 and f3. In general, a zero of fp is also a zero of fpq, because a p-cycle is also a pq-cycle.

The other zero of f2 is the center of the 2-cycle disc.

The other zero of f3 is the center of the 3-cycle cardioid.

On the right we see the graphs of f3, f6, and f9, near the left zero of f3.

Note this plot resembles a vertical reflection fo the left plot.

This is the key observation of renormalizaion: rescaled higher polynomials resemble the original polynomials.

Return to Hurwitz-Robucci scaling.