Rational maps with Fatou components of arbitrarily large connectivity

Abstract

We study the family of singular perturbations of Blaschke products B a,λ (z) = z 3 z−a 1−az + λ z 2 . 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 B a,λ which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.

Achievable connectivities of Fatou components for a 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 that case, if c − (λ) belongs to a Fatou component in A(∞) which surrounds z = 0, the dynamical plane has Fatou components of arbitrarily large finite connectivity. The actual existence of parameters for which this actually happens was proven in [Can18]. We want to point out that the same results can be proven for n, d ≥ 2 such that 1/n + 1/d < 1.…”

“…Moreover, for any given n ∈ N, there are examples of rational maps with Fatou components of connectivity n. These examples can either be obtained by quasiconformal surgery (see [BKL91]) or by giving explicit families of rational maps (see [QG04] and [Ste93]). 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].…”

“…Moreover, if a ∈ D * the map B n,a has only two simple critical points c − ∈ A * (0) and c + ∈ A * (∞), other than the superatracting fixed points z = 0 and z = ∞. In [Can17,Can18] the author studied the family of singular perturbations of the maps B n,a given by…”