The escape trichotomy for singularly perturbed rational maps

In this paper we consider the dynamical behavior of the family of complex rational maps given by F-lambda (z) = z(n) + lambda/z(d) where n >= 2, d >= 1. Despite the high degree of these maps, there is only one free critical orbit up to symmetry. Also, the point at infinity is always a superattracting fixed point. Our goal is to consider what happens when the free critical orbit tends to infinity. We show that there are three very different types of Julia sets that occur in this case. Suppose the free critical orbit enters the immediate basin of attraction of infinity at iteration j. Then we show: (1) If j = 1, the Julia set is a Cantor set; (2) If j = 2, the Julia set is a Cantor set of simple closed curves; (3) If j > 2, the Julia set is a Sterpinski curve.