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.
Categories: math.DS