Fatou components and Julia sets of singularly perturbed rational maps with positive parameter

In this paper, we discuss the rational maps $F_\lambda (z) = z^n + \lambda /z^n ,n \geqslant 2$ with the positive real parameter .. It is shown that the immediately attracting basin B . of . for F . is always a Jordan domain if the Julia set of F . is not a Cantor set. Furthermore, B . is a quasidisk if there is no parabolic fixed point on the boundary of B . . It is also shown that if the Julia set of F . is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpi.ski curve is given.