In the iteration of rational maps, given a map f , can we modify f to obtain a new map with richer dynamics? When f has a superattracting periodic orbit, we may insert the Julia set of a quadratic polynomial P into an attracting basin.
* This is called a tuning.
* When there are two independent superattracting orbits, we may insert two polynomials P and Q ; this is a mating.
* If there are two critical points in the same orbit, we may insert the Julia sets of two quartic polynomials P°Q and Q°P ; this is an anti-mating.
By varying P and Q , there are related maps from polynomial parameter spaces (for example, the Mandelbrot set) to rational parameter spaces, which are well-understood only in the case of tuning. For mating and anti-mating there are only partial results, in particular for postcritically finite maps.
