Singular perturbations of Blaschke products and connectivity of Fatou components

Jordi Canela

doi:10.3934/dcds.2017153

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.

Highlights

Given a Rational map f : C → C, where C denotes the Riemann sphere, the Fatou set F(f ) is defined as the set of points z ∈ C such that the family of iterates {f (z), f 2(z) = f (f (z)), ...} is normal in some open neighbourhood of z
The aim of this paper is to study singular perturbations of a family of Blaschke products and analyse the structure of their dynamical plane
In this paper we introduce a family of rational maps with McMullen-like Julia sets which present different dynamics than the one of the previously mentioned works

Summary

Introduction

We consider singular perturbations of rational maps whose Julia set is the unit circle and which have free critical points, i.e. critical points which do not belong to a superattracting cycle The existence of these free critical points allows the appearance of other types of dynamics in the Fatou set. In Theorem A we prove that, fixed a ∈ D∗, if λ ∈ C∗ = C \ {0}, |λ| is mall enough, and c−(a, λ) belongs to the basin of attraction of z = ∞, A(∞), there are only three possibilities: either every Fatou component has connectivity less or equal than two, or less or equal than three, or one can find components of arbitrarily large finite connectivity This family of degree 6 rational functions may contain maps with preperiodic.

Preliminaries

Fatou set of the singularly perturbed Blaschke products: proof of Theorem A

Julia set of the singularly perturbed Blaschke products: proof of Theorem B