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 right-click to save the file to your hard disk, and open it with a video player. Clicking show opens a small pop-up box to play the video in reduced size.

Bitransitive rational maps

For quadratic rational maps fa(z) = 1 + a/z2, the critical point 0 is mapped to the critical point ∞, and every hyperbolic map is bitransitive. 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.

Mating and repelling capture

While a hyperbolic mating is never bitransitive, preperiodic maps fa(z) may be matings in fact. The simplest examples are of the form θ ∐ -2θ with a direct ray connection; here θ has preperiod at least two. When -2θ converges to a periodic angle, the geometric mating probably converges to a parabolic capture, which is not the mating with the limiting polynomial.

phase space video for Lattès 224

Does the map θ ↦ fa ∼ θ ∐ -2θ of preperiodic angles extend continuously? See it as a deformation of M: play — show.
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Anti-mating

Anti-mating was introduced by Ahmadi, Timorin, and Meyer. Consider two polynomials acting between two planes, P(z) = z2 + p from the first plane to the second plane, and Q(z) = z2 + q in the opposite direction. Mapping these planes to half-spheres defines the formal anti-mating P ⊓ Q. The topological anti-mating P ∏ Q is obtained by gluing the filled Julia sets of Q°P and P°Q along their boundaries. For q = 0 we have Q°P(z) = z4 + q and P°Q(z) = (z2 + q)2. This anti-mating is never obstructed, and defines some fa(z) = 1 + a/z2 ∼ P ∏ Q. See the forthcoming paper [B4].

phase space videos for 224 and for hyperbolic maps are under construction ...

Slow deformation of M4 in parameter space: play — show.
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Generalized anti-mating

The limbs of the period-2 component, which result from anti-mating, form only part of parameter space, and a generalized construction is needed for higher periods. See also the approach of Pascale Roesch and Bastien Rossetti, which employs parabolic surgery for molecule parameters.