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 (or escaping). 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

Two postcritically finite quadratic polynomials P(z) and Q(z) are combined to define the formal mating P ⊔ Q: in the lower and upper half-spheres, this new map is non-analytically conjugate to P(z) or Q(z), respectively. The Thurston algorithm gives an equivalent rational map f(z), if P(z) and Q(z) do not belong to conjugate limbs of the Mandelbrot set. It is conjugate to the topological mating P ∐ Q, where the filled Julia sets are glued together. 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.

Nevertheless, the map θ ↦ fa ∼ θ ∐ -2θ of preperiodic angles might extend continuously. See it as a deformation of M: play — show.
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The Lattès map f-2(z) = 1 - 2/z2 of type (2, 4, 4) is a mating 1/4 ∐ 1/2 of this form; on the left there is a corresponding anti-mating, see below: play — show.
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Two more (rotated) representations of f-2(z) are given by 5/12 ∐ 1/6 and 13/28 ∐ 3/14: 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 p = 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 [B8].

There are anti-matings of real polynomials in this family; they are not obstructed, because the rays with angles ± 1/3 never land together. q3 = -2 gives the Lattès map of type (2, 4, 4) discussed above, and q3 = -1 gives a 4-periodic hyperbolic map: play — show.
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Two more hyperbolic examples of period 6: play — show.
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Slow deformation of the (inverted) Multibrot set 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.