Rational maps with Fatou components of arbitrarily large connectivity

We study the family of singular perturbations of Blaschke products Ba,λ(z)=z3z−a1−a‾z+λz2. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter λ. We prove that all possible escaping configurations of the critical point c−(a,λ) take place within the parameter space. In particular, we prove that there are maps Ba,λ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.