Abstract:

I am going the present a family of rational maps called "singular perturbation". The family is a perturbation of the map $z to  z^n$ adding a pole of order d>0 at z=0. Even tough the degree of the map can be arbitrarily high, the symmtries due to the concrete formula implies that the family has a unique free singular orbit. This family has been widely studied in the last two decades by many people. I will describe the basic notions and I will state and give some hints of the proofs of the Trichotomy Theorem which partially describe the topology of the Julia and Fotou sets under the assumption of all singular points escaping to infinity.