Connectivity Of Fatou Sets And Julia Sets

The study on the topological properties of the Fatou sets and Julia sets of the rational functions and holomorphic functions has formed rather mature methods and contents.But the dynamics of non-holomorphic functions is not yet fully clear and still need further study.Gomez and Lopez de Medramo studied the classification of dynamics of maps of degree two.When they studied whether the dynamical behaviors of the holomorphic functions can be extended to the dynamical behaviors of the non-holomorphic functions,they mentioned the family f_?=z~2+?(?).The primary goal of this paper is to prove that the attracting domain at the origin is simply connected.Firstly,we recall some classical results of dynamics of the rational functions and holomorphic functions,including the theories on the Julia sets and Fatou sets.Moreover,we state the results of connectivities of Fatou sets and Julia sets of rational functions through the examples of polynomials and Mcmullen maps.Then,we give some basic properties of the family f_?=z~2+?(?).For example,the set of critical points is a circle at the origin whose radius is |?|/2.The distribution of the pre-images is discussed according to the distribution of the critical values.At last,we prove that the set of critical values is a Jordan curve and the conclusion also holds for all all the pre-images of f_?(z).Based on this,we get that the attracting domain of f_?=z~2+?(?),??(0,2/3)at the origin is simply connected.