Rational maps with a superattracting 3-cycle:Quadratic rational maps fa(z) = (z2 + A) / (z2 - a2) with A = a3 - a - 1 form a family denoted by V3 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–Chéritat approach to slow mating and tuning.
MatingTwo 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) = z2 - 1.754878 is the airplane polynomial and P(z) does not belong to the 1/2-limb, or Q(z) = z2 - 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 fa(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.
CaptureStarting 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 ∞.
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