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

Abstract

In this paper, we discuss the rational maps $$F_\lambda (z) = z^n + \lambda /z^n ,n \geqslant 2$$ with the positive real parameter λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Furthermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ 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 Sierpinski curve is given.