Singular Perturbations with Multiple Poles of the Simple Polynomials

Abstract

In this article, we study the dynamics of the following family of rational maps with one parameter:where n ≥ 3 and λ ∈ C * . This family of rational maps can be viewed as a singular perturbations of the simple polynomial P n (z) = z n . We give a characterization of the topological properties of the Julia sets of the family f λ according to the dynamical behaviors of the orbits of the free critical points.

Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

“…It is known that the Cantor circle Julia sets and Sierpiński carpet Julia sets can appear in McMullen family and the generalized McMullen family (see [5,30]). For the study of singular perturbation of unicritical polynomials, one may also refer to [27], [31], [32], [15] and [16].…”