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## Rational maps with a superattracting 3-cycle

Quadratic rational maps f_{c}(z) = (z

^{2}+ A) / (c

^{2}- z

^{2}) with A = c

^{3}- c - 1 form a family denoted by V

_{3}, since the critical point ∞ is 3-periodic. The videos illustrate applications of the Thurston algorithm to construct these maps from polynomials. The visualization of mating is based on the Buff–ChÃ©ritat 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, the new map is non-analytically conjugate to P(z) or Q(z), respectively. The Thurston algorithm gives a rational map, if P(z) and Q(z) do not belong to conjugate limbs of the Mandelbrot set. In particular, if Q(z) = z^{2}- 1.754878 is the Airplane polynomial and P(z) does not belong to the 1/2-limb, or Q(z) = z

^{2}- 0.662359 + 0.562280 i is the Rabbit polynomial and P(z) does not belong to the 2/3-limb, then the resulting map will be of the form f

_{c}(z) . It is conjugate to the topological mating P ∐ Q, where the filled Julia sets are glued together.

A simple example of a shared mating, Airplane ∐ Rabbit is affine conjugate to Rabbit ∐ Airplane: play — show.

(The red and blue curves show convergence of marked points from slow mating.)

More phase space videos, also with moving Julia sets, are under construction ...

The slow deformation from M to M ∐ Rabbit and M ∐ Airplane: play — show.

The locus of P in the 1/2-limb mated with the Rabbit: play — show.

The locus of P in the 1/3-limb mated with the Airplane: play — show.

On the left M_1/2 ∐ R, properly contains M_1/3 ∐ A on the right: play — show.

The 1/2-vein mated with R, and the 1/4-vein mated with A: play — show.

Airplane matings, on the left the limbs with 2/7 < θ < 1/3, and on the right 0 < θ < 1/7: play — show.

While the upper left part of the non-escape locus probably can be described as a mating of the Mandelbrot set with the Rabbit, this is not obvious for the Airplane. See the location of the Airplane spine, with 1/7 ≤ θ ≤ 3/14 on the left and 2/7 ≤ θ < 1/3 on the right: play — show.

### Generalized anti-mating

Under construction ... see also this presentation.### Hyperbolic capture and regluing

Starting with the Airplane or Rabbit polynomial, move the critical value ∞ to a preimage of the 3-periodic critical point 0, preferably along external and internal rays. This process will be shown inverted, with 0 moved to the Airplane or Rabbit around ∞.Phase space videos under construction ...

Zooms of the mating locus above show capture components, such that only half of the boundary consists of matings. On the right, only the 3/8-limb is shown: play — show.

This answers a question by Rees, Mashanova, and Timorin. See notes on the interpretation

X
The videos of the mating locus and the deformation videos are made by applying
the slow mating algorithm to a large number of Misiurewicz parameters of a high
preperiod:

* If the red curve surrounds a small Mandelbrot set in some uniform distance, this will be due to the limited preperiod, and probably the small Mandelbrot set belongs to the mating locus;

* If a more irregular open set containing many small Mandelbrot sets is not red, it will probably be disjoint from the mating locus;

* if the red line crosses the root of a satellite component, probably the satellite Mandelbrot set will belong to the mating locus but the bigger one does not;

* At parabolic parameters, there are fewer points and hence red line segments will be longer. Also, convergence will be slower.

* If the red curve surrounds a small Mandelbrot set in some uniform distance, this will be due to the limited preperiod, and probably the small Mandelbrot set belongs to the mating locus;

* If a more irregular open set containing many small Mandelbrot sets is not red, it will probably be disjoint from the mating locus;

* if the red line crosses the root of a satellite component, probably the satellite Mandelbrot set will belong to the mating locus but the bigger one does not;

* At parabolic parameters, there are fewer points and hence red line segments will be longer. Also, convergence will be slower.

X
Fatou components on the Airplane spine can be accessed from two sides. Since
the rays are not homotopic, this may give different maps of type C and different
capture components. Then only half of the boundary consists of matings related
to the boundary of this Fatou component. It was suggested that the same
hyperbolic component may be a capture from another Fatou component, but in the
example of c ≈ 0.6930 + 0.1847 i, there is no other candidate with
f

_{c}^{5}(0)=-c and angles between 2/7 and 1/3. Note that most of this argument is combinatorial; the numerical part is only in the location in the upper halfplane, so for rays from the other side, there will be another component in the lower halfplane.See also the 3/8- and 2/5-limbs deform into the Airplane mating locus: play — show.

And there are indirect matings, with longer ray connections, on part of the boundary of a capture component as well; a simple example is 6999/32704 ∐ A: play — show.