Singular perturbations of zn with multiple poles

We study the dynamics of the family of complex maps given by f lambda(z) = z(n) + lambda/((z - a)(da)(z - b)(db)) where n >= 2 is an integer and lambda is an arbitrarily small complex parameter. We focus on the topological characteristics of the Julia set and the Fatou set of f(lambda)(z). We prove that despite the large amount of possibilities there are only four different cases that correspond to di. erent positions and orders of the poles a and b.