Singular perturbations with multiple poles of the simple polynomials

TL;DR

This paper investigates the complex dynamics of a family of rational maps that are singular perturbations of simple polynomials, focusing on the topological properties of their Julia sets based on critical orbit behavior.

Contribution

It provides a characterization of Julia set topologies for the family of rational maps with multiple poles, extending understanding of singular perturbations in complex dynamics.

Findings

Julia set topology varies with critical orbit behavior

Critical points determine the dynamical regimes of the maps

The family exhibits rich dynamical structures due to singular perturbations

Abstract

In this article, we study the dynamics of the following family of rational maps with one parameter: \begin{equation*} f_\lambda(z)= z^n+\frac{\lambda^2}{z^n-\lambda}, \end{equation*} where $n\geq 3$ and $\lambda\in\mathbb{C}^*$. 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 $f_\lambda$ according to the dynamical behaviors of the orbits of the free critical points.