Rational maps with Fatou components of arbitrarily large connectivity

We study the family of singular perturbations of Blaschke products $B_{a,\lambda}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{\lambda}{z^2}$. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter $\lambda$. We prove that all possible escaping configurations of the critical point $c_-(a,\lambda)$ take place within the parameter space. In particular, we prove that there are maps $B_{a,\lambda}$ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.