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Rational maps with Fatou components of arbitrarily large connectivity (1704.00544v1)
Published 3 Apr 2017 in math.DS
Abstract: We study the family of singular perturbations of Blaschke products $B_{a,\lambda}(z)=z3\frac{z-a}{1-\overline{a}z}+\frac{\lambda}{z2}$. 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.