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.

Symmetric rational maps

Quadratic rational maps fc(z) = (z2 + c) / (1 + cz2) form the family of symmetric maps, since they commute with the inversion z ↦ 1/z. 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 anti-mating

Definitions of mating and anti-mating are given here.
Two symmetric maps are related by f1/c(z) = fc(1/z) = 1/fc(z), and we have fc(z) ∼ P ∐ P, if and only if f1/c(z) ∼ P ∏ P.
Mixung is a generalization of self-anti-matings, see arXiv:2209.08012.

The self-matings of the Rabbit and of the Kokopelli: play — show.
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The classical example of a shared mating, Airplane ∐ Rabbit is conjugate to Rabbit ∐ Airplane by a rotation. They are conjugate to a symmetric map fc(z), which is not a self-mating: play — show.
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Deformation of M to the loci of self-matings and and self-anti-matings: play — show.
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Locus of the 1/3-limb mated and anti-mated with itself: play — show.
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The 1/4-vein mated and anti-mated with itself: play — show.
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Lattès maps

When a mating gives a Lattès map of type (2, 2, 2, 2), the Thurston pullback diverges in general; the point configurations spiral toward a center manifold. This is illustrated with a zoom of the video around a postcritical point. See the joint paper [B6] with Arnaud Chéritat and the poster from the conference celebrating Jack Milnor.

1/12 ∐ 5/12 (divergent, non-self, f not symmetric): play — show.
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See the path of point configurations: play — show.
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1/4 ∐ 1/4 (convergent, self-mating, f symmetric): play — show.
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See the path of point configurations: play — show.
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53/60 ∐ 29/60 (divergent, non-self, f symmetric, the same): play — show.
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See the path of point configurations: play — show.
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3/14 ∐ 3/14 (divergent, self-mating, f symmetric): play — show.
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See the path of point configurations: play — show.
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1/6 ∐ 1/6 (divergent, self-mating, f symmetric): play — show.
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See the path of point configurations: play — show.
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Chebyshev maps

For a Chebyshev map ya(z) = (-z2 + a + 2) / (z2 + a), we have ∞ ↦ -1 ↦ 1, and 1 is fixed. The Petersen transformation sends fc(z) and f1/c(z) to the same Chebyshev map, and especially P ∐ P and P ∏ P to P ∐ T with T(z) = z2 - 2.

other possibilities, here veins through period 3, when equal

veins ...: play — show.
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veins ...: play — show.
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