IUMJ

Title: The escape trichotomy for singularly perturbed rational maps

Authors: Robert L. Devaney, Daniel M. Look and David Uminsky

Issue: Volume 54 (2005), Issue 6, 1621-1634

Abstract:

In this paper we consider the dynamical behavior of the family of complex rational maps given by $F_{\lambda}(z) = z^n + \frac{\lambda}{z^d}$ where $n \geq 2$, $d \geq 1$. Despite the high degree of these maps, there is only one free critical orbit up to symmetry. Also, the point at $\infty$ is always a superattracting fixed point. Our goal is to consider what happens when the free critical orbit tends to $\infty$. 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 $\infty$ 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.