Polynomial matings on the Riemann sphere
Welcome. I did not work out thumbnails yet. Here's a little guide throughout the directory structure of MatMovies.
You'll find in these directories various animations. You may like to browse them according to the following list.
Ray numbers: they indicate the argument of external rays tending to the parameter in the Miziurewicz case ( <=> strictly preperiodic critical point => subhyperbolic polynomial and dendrite Julia set). In the hyperbolic case ( <=> periodic critical point => Julia set enclosing bounded components of Fatou set ) there are two external rays indicated, and they land not on the parameter but on another one, which is the root of the hyperbolic component containing the correct parameter.
- RabbitBasilica = mating of Douady's rabbit (ray=1/7 and 2/7) with the basilica (ray=1/3 and 2/3). The original example of mating.
- RabbitRabbit = mating of Douady's rabbit (ray=1/7 and 2/7) with another Douady's rabbit.
- RabbitPrim4 = mating of the Rabbit (1/7,2/7) with a period 4 primitive example in the 1/3 limb (ray=3/15 and 4/15).
- AirplaneRabbit = the Airplane (3/7,4/7) with the rabbit (1/7,2/7). This one has longer ray connexions, leading to a limit where it is hard to recognize the two original Julia sets.
- DendriteDendrite = a miziurewicz (ray=1/4) mated with an identical copy (ray=1/4 again). The limit is a Lattes example, it has a "regular" invariant measure (it is kind of evenly distributed) and is moreover related to the dragon area filling fractal curve. This is an example of Milnor.
- DendriteOtherdendrite If one mates 1/4 with 1/6 instead, the limit looks much less regular.
- DendriteRabbit = The 1/6 Mizurewicz with (1/7,2/7).
- AirplanePrim4 = Airplaine (3/7,4/7) with period 4 primitive copy in the 1/3 limb (3/15,4/15). See how intricate this gets. This is another example of shared mating.
- ReesSharedExamples = an example of shared matings.
- Deg3Newton = Mating two particular degree 3 polynomials leads to the Julia set of a Newton method of another degree 3 polynomial (communicated by Tan Lei).
- Siegels = Mating Siegel disks.
- Tunings = Polynomial tuning is a close relative of polynomial mating.
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