Singular perturbations of complex polynomials

Abstract

Abstract. 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.

A Mandelpinski maze for rational maps of the form zn+λ/zd

“…For more details about the dynamical properties of these maps and the structure of the parameter plane, see [4].…”

Exotic topology in complex dynamics

“…For simplicity, we will restrict in this paper to the case where n = d = 2, though much of what happens in this case occurs in the more general family. See [5] for more details about the general family. Curiously, as we briefly discuss below, the case where n = d = 2 is the most complicated of these families [4].…”

“…It is known that the union of these sets forms the Fatou set for F; the Julia set is its complement. See [10,5] for more details.…”

Singular Perturbations with Multiple Poles of the Simple Polynomials

“…where P(z) is a polynomial with degree n ≥ 2 whose dynamics are completely understood, a ∈ C, d ≥ 1 and λ ∈ C * [5]. When P(z) = z n with n ≥ 2, a = 0, d ≥ 2 and λ ∈ C * , the family of rational maps F λ is commonly called the McMullen maps, which has been studied extensively by Devaney and his collaborators in a series of articles (see [3,5,7,8]).…”

“…When P(z) = z n with n ≥ 2, a = 0, d ≥ 2 and λ ∈ C * , the family of rational maps F λ is commonly called the McMullen maps, which has been studied extensively by Devaney and his collaborators in a series of articles (see [3,5,7,8]). Specifically, it is proved in [7] that if the orbits of the critical points of F λ are all attracted to ∞, then the Julia set of F λ is either a Cantor set, a Sierpiński curve, or a Cantor set of circles.…”