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 f_p is also a zero of f_pq, 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.