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 R(z, t) that A. F. Beardon proposed, we prove that R(z, t) has Fatou components with connectivities 3 and 5 for any t epsilon (0, 1/12]. Furthermore, there exists t epsilon (0, 1/12] such that R(z, t) has Fatou components with connectivity nine.