## Rational maps with a superattracting 3-cycle:

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

^{2}+ A) / (z

^{2}- a

^{2}) with A = a

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

_{3}since the critical value ∞ 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 and tuning.

### Mating

Two postcritically finite quadratic polynomials P(z) and Q(z) are combined to define the formal mating: in the lower and upper half-spheres, the new map is non-analytically conjugate to P(z) or Q(z), respectively. The (modified) 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

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

A simple example of a shared mating, airplane & rabbit coincides with rabbit & airplane: play — show.

Two mating loci together: play — show.

### Capture

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 ∞.Under construction ...