Singular Perturbations of Complex Polynomials

In this paper we describe the dynamics of singularly perturbed complex polynomials. That is, we start with a complex polynomial whose dynamics are well understood. Then we perturb this map by adding a pole, i.e., by adding in a term of the form λ/(z − a)d where the parameter λ is complex. This changes the polynomial into a rational map of higher degree and, as we shall see, the dynamical behavior explodes. One aim of this paper is to give a survey of the many different topological structures that arise in the dynamical and parameter planes for these singularly perturbed maps. We shall show how Sierpiński curves arise in a myriad of different ways as the Julia sets for these singularly perturbed maps, and while these sets are always the same topologically, the dynamical behavior on them is often quite different. We shall also describe a number of interesting topological objects that arise in the parameter plane (the λ-plane) for these maps. These include Mandelpinski necklaces, Cantor webs, and Cantor sets of circles of Sierpiński curve Julia sets.