Polynomial matings on the Riemann sphere

A degre 3 non-matable example with a non-Levy obstruction

Silvio Levy proved that Thurston obstructions of degree 2 post-critically finite ramified covers are necessarily of a simple form now called Levy obstructions. Tan Lei used Levy's criterion to give a very neat characterization of obstructed formal matings of post-critically finite polynomials of degree 2: there is an obstruction if and only if they belong to conjugate limbs of the Mandelbrot set.

In higher degree the situation is more complex. Tan Lei and Shishikura presented and studied in a joint work an example of formal mating with a non-Levy obstruction.

I made a movie of the slow mating procedure for their example. There are a lot of interesting phenomena visible, several of which are now explained.

You can also download:

• The movie above (mp4, 1.7MB)
• The movie above (mov, 1.7MB)
• Variant (mov, 2.3MB) normalized to zoom on the middle of a "tube".
• variant (mov, 4.9MB) normalized to zoom on another part of a tube.
• variant (mov, 7.1MB) different color scheme, higher quality, normalized on the middle sphere, non-symmetric.