Polynomial matings on the Riemann sphere
Rabbit with BasilicaThis is the historical example.
The picture above is part of a numerical experient carried-out by Douady and Hubbard in the 1980's. They were computing rough pictures of the Julia sets of all post critically finite degree 2 rational maps with a given bound on the size of post critical set. Looking at the picture above Douady and Hubbard recognized the gluing of the filled in Julia sets of two post-critically finite degree polynomials they were quite familiar with; they had even given them nicknames: the Basilica and the Rabbit.
And above you can see a more detailed picture of the Julia set, drawn on the complex plane. The rational map has two critical points, 0 and infinty. The point 0 has period 2 and infinity has period 3. Their orbits are indicated by red dots (the one at infinity is not visible).
Since then, it has been proved in a precise way that the Julia set is indeed the mating of the basilica and the rabbit.
Below: the so-called slow mating.
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