On Connectivity of Fatou Components concerning a Family of Rational Maps

Abstract

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