Séminaire de Systèmes Dynamiques

Tuning, mating, and anti-mating for quadratic rational maps

by Wolf Jung

207 (Bat 1R2)


Bat 1R2


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.