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 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 oo is always a superattracting fixed point. Our goal is to consider what happens when the free critical orbit tends to ∞. 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 ∞ 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 Sierpinski curve.