The escape trichotomy for singularly perturbed rational maps Robert DevaneyDaniel LookDavid Uminsky 37F1037F45Julia setrational mapescape trichotomyMcMullen DomainSierpinski curve 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2615 10.1512/iumj.2005.54.2615 en Indiana Univ. Math. J. 54 (2005) 1621 - 1634 state-of-the-art mathematics http://iumj.org/access/