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.
Categories: math.DS