# Singular perturbations of Blaschke Products and connectivity of Fatou components

## TL;DR

This paper investigates how singular perturbations affect the connectivity and structure of Fatou components and Julia sets in a family of rational maps derived from Blaschke products, revealing complex topological behaviors.

## Contribution

It demonstrates that singular perturbations can produce Fatou components with arbitrarily large connectivity and a Julia set composed of Cantor sets of quasicircles and points, under specific conditions.

## Key findings

- Fatou components can have arbitrarily large finite connectivity.
- Julia set consists of countably many Cantor sets of quasicircles and uncountably many points.
- Under certain conditions, all Fatou components have finite connectivity.

## Abstract

The goal of this paper is to study the family of singular perturbations of Blaschke products given by $B_{a,\lambda}(z)=z^3\frac{z-a}{1-\overline{a}z}+\frac{\lambda}{z^2}$. We focus on the study of these rational maps for parameters $a$ in the punctured disk $\mathbb{D}^*$ and $|\lambda|$ small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product $B_{a,\lambda}$ have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.