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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...
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Citations
... We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [Can17,Can18]. ...
... However, the degree of the rational maps obtained in all previous examples grows rapidly with n. To our knowledge, the first example of rational map whose dynamical plane contains Fatou components of arbitrarily large finite connectivities was presented in [Can17] (see also [Can18]) by using singular perturbations. However, in these papers it is not shown which precise connectivities can actually be attained. ...
... However, in these papers it is not shown which precise connectivities can actually be attained. The goal of this paper is to study the attainable connectivities for a wider family of singular perturbations which includes the ones studied in [Can17,Can18]. We also want to remark that while this paper was being prepared we knew that, independently, professor Hiroyuki has obtained another family of rational maps with Fatou components of arbitrarily large connectivity [Hir]. ...
In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [Can17, Can18].
In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [5,6].
