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: \begin{equation*} f_λ(z)= z^n+\frac{λ^2}{z^n-λ}, \end{equation*} where $n\ge 3$ and $\lambda \in {\mathbb{C}}^{\ast }$. 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 $f_{\lambda }$ according to the dynamical behaviors of the orbits of the free critical points.

15 pages, 5 figures, to appear in Qualitative Theory of Dynamical Systems