  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.
