Article Contents
Article Contents

# Achievable connectivities of Fatou components for a family of singular perturbations

• Corresponding author: Dan Paraschiv
• 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].

Mathematics Subject Classification: Primary: 37F10, 37F12, 37F20, 37F44; Secondary: 30D05.

 Citation:

• Figure 1.  Dynamical planes of the family $B_{n, d, a, \lambda}(z)$ for $d = 3$. The top-left figure corresponds to $n = 2$ and $\lambda = 2\cdot10^{-8}$; the top-right corresponds to $n = 3$ and $\lambda = -5\cdot10^{-8}$; the bottom-left figure corresponds to $n = 4$ and $\lambda = -6.3\cdot10^{-9}$; and bottom-right corresponds to $n = 5$ and $\lambda = -1.2\cdot10^{-10}$. In all cases we can see the triply connected regions (where the critical point $\nu_\lambda$ lies) and their eventual preimages, which are Fatou components with increasing connectivity

Figure 2.  Left figure illustrates the dynamical planes of $M_{n, a}$ for $n = 2$ and $a = (0.9+0.6i)$. Right picture illustrates the dynamical plane of the (perturbed) family $\mathcal{S}_{n, d, \lambda}$ when the unperturbed map is precisely $M_{2, a}$, and the pertubation corresponds to $d = 3$ and $\lambda = -10^{-7}$. We can see in the right figure the triply connected Fatou component which contains $\nu_{ \lambda}$ and its eventual preimages with higher connectivity

Figure 3.  Partition of the dynamical plane with respect to $\mathcal{A}^{*}_{ \lambda}(\infty)$, $A_{ \lambda}$, $T_{ \lambda}$, and $D_{ \lambda}$, described in Proposition 2. Blue and purple points denote zeros and critical points, respectively

Figure 4.  Partitions of the dynamical plane introduced in Theorem 3.5

Figure 5.  Description of the situation in the proof of Lemma 4.2, where $k = 2$ and $W_2 \subset \mathcal{U}_d$. In this case, $B_2 = B_{2}^{ \text{out}} \cup \overline{ A_{ \lambda}} \cup B_{2}^{ \text{in}}$

Figure 6.  The top figures correspond to the possible cases of $\nu_{ \lambda}$ lying in a neighbourhood of $\partial^ \text{Int} A_{m, \lambda}$. The top figures correspond to the possible cases of $\nu_{ \lambda}$ lying in a neighbourhood of $\partial^ \text{Ext} A_{m, \lambda}$

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