The video clips illustrating complex dynamics typically have a few MB and a duration of about 30s. They were produced with FFmpeg, combining images made with Mandel. For each video, two links are given: clicking play will open the video in a new browser tab or window. Or click right to save the file to your hard disk, and open it with a video player, e.g., VLC. This is recommended for large file size in particular. Clicking show shall open a small pop-up box to play the video, possibly in reduced size. (When you are closing the pop-up before the video is completed, please stop playback first.)

Rational maps with a superattracting 3-cycle:

Quadratic rational maps fa(z) = (z2 + A) / (z2 - a2) with A = a3 - a - 1 form a family denoted by V3 since the critical value ∞ is 3-periodic. The videos illustrate applications of the Thurston algorithm to construct these maps from polynomials. The visualization of mating is based on the Buff–Chéritat approach to slow mating and tuning.

Mating

Two postcritically finite quadratic polynomials P(z) and Q(z) are combined to define the formal mating: in the lower and upper half-spheres, the new map is non-analytically conjugate to P(z) or Q(z), respectively. The (modified) Thurston algorithm gives a rational map, if P(z) and Q(z) do not belong to conjugate limbs of the Mandelbrot set. In particular, if Q(z) = z2 - 1.754878 is the airplane polynomial and P(z) does not belong to the 1/2-limb, or Q(z) = z2 - 0.662359 + 0.562280 i is the rabbit polynomial and P(z) does not belong to the 2/3-limb, then the resulting map will be of the form fa(z) . It is conjugate to the topological mating of P(z) and Q(z), where the filled Julia sets are glued together.

A simple example of a shared mating, airplane & rabbit coincides with rabbit & airplane: play — show.
X


Two mating loci together: play — show.
X

Capture

Starting with the airplane or rabbit polynomial, move the critical value ∞ to a preimage of the 3-periodic critical point 0, preferably along external and internal rays. This process will be shown inverted, with 0 moved to the airplane or rabbit around ∞.

Under construction ...