Examples of Simply and Multiply Connected Fatou Sets for a Class of Meromorphic Functions

Abstract

Aims: We give some families which are meromorhic outside a compact countable set B of essential singularities. Our aim is to give some examples of the stable set (called the Fatou set) and the unstable set (called the Julia set) since there is not study of examples of any parametric family of this class of functions (called in the introduction functions of class K) in complex dynamics. Study design: We study components of the Fatou set and some theorems related with the iteration of functions in class K and design a computational program to give examples of the Julia and Fatou sets. Place and Duration of Study: F.C. F´isico-Matem´ ticas, Benem´ erita Universidad Aut ´onoma de Puebla, M´exico between June 2011 and July 2012. Methodology: We use some theorems of complex dynamics in order to study components of the Fatou set. We program some algorithms in C and get the picture of this set. Results: Given a family in class K we get some mathematical results of the Fatou and Julia sets and its pictures for some parameters given. Conclusion: Taking some families in class K ∩ Sk we give examples of the Fatou set which can be either simply-connected or multiply-connected in the last case the Julia set is a totally disconnected set.