Singular perturbations of Blaschke Products and connectivity of Fatou components

The goal of this paper is to study the family of singular perturbations of Blaschke products given by $B_{a,\lambda}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{\lambda}{z^2}$. We focus on the study of these rational maps for parameters $a$ in the punctured disk $\mathbb{D}^*$ and $|\lambda|$ small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product $B_{a,\lambda}$ have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.