Singular Perturbations in the McMullen Domain

In this paper we study the dynamics of the family of complex maps given by fλ(z) = z n + λ/(zh(z − a)d) where n, h and d are integers such that 1/n+1/h < 1 and d ≥ 1 and the pole a lies outside the unit circle. The parameter λ is complex and arbitrarily small. In spite of the large number of possibilities we prove that the Julia set of fλ(z) consists of a countable collection of Cantor sets of circles only one of which surrounds the origin and uncountably many point components that accumulate on each of these curves. ∗Please address all correspondence to Robert L. Devaney, Department of Mathematics, Boston University, 111 Cummington St., Boston, MA 02215, USA. Email address: [email protected].