Singular Perturbations with Multiple Poles of the Simple Polynomials

In this article, we study the dynamics of the following family of rational maps with one parameter: f λ ( z ) = z n + λ 2 z n - λ , 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.