Polynomial matings on the Riemann sphereHere you will find movies of polynomial matings on the Riemann sphere.
Polynomial mating is the name given by Douady an Hubbard to a phenomenon that they discovered: the Julia sets of some rational maps is homeomorphic to the gluing of two filled-in polynomial Julia sets along their boundaries, by a gluing that respects the marking by external angle and thus the corresponding dynamical systems on the Julia sets.
A progressive interpolation was introduced, between the two polynomial Julia sets and their mating. It consists in gluing equipotentials together and gives a holomorphic dynamical system between different spheres (this was observed by Milnor). This dynamical systems gives an easy method for drawing a conformally correct picture of the deformation of the polynomial Julia sets under the equipotential gluing: this method was explained to me by Buff. The result is an image which depends on the potential. This is what the movies show: the potential starts high and slowly approaches 0.