Achievable connectivities of Fatou components for a family of singular perturbations

TL;DR

This paper investigates the connectivity properties of Fatou components in a family of singular perturbation maps, demonstrating the existence of components with arbitrarily large connectivity and precisely characterizing these connectivities.

Contribution

It extends previous results by precisely determining the connectivities of Fatou components for certain parameters within the family of singular perturbation maps.

Findings

Existence of Fatou components with arbitrarily large connectivity

Precise characterization of these connectivities

Extension of previous connectivity results

Abstract

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].