On Connectivity of Fatou Components concerning a Family of Rational Maps

I. N. Baker established the existence of Fatou component with any given finite connectivity by the method of quasi-conformal surgery. M. Shishikura suggested giving an explicit rational map which has a Fatou component with finite connectivity greater than 2. In this paper, considering a family of rational maps Rz,t that A. F. Beardon proposed, we prove that Rz,t has Fatou components with connectivities 3 and 5 for any t∈0,1/12. Furthermore, there exists t∈0,1/12 such that Rz,t has Fatou components with connectivity nine.