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

Aims: We give some families of functions which are meromorphic 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 are not many examples of 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: Facultad de Ciencias Fsico Matematicas, Benemerita Universidad
Autonoma de Puebla, Mexico 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 some gures of the Fatou set.
Results: Given a family of functions in class K we get some mathematical results of the Fatou and
Julia sets and its gures 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.