On a Family of Rational Perturbations of the Doubling Map

Abstract

The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$. First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter $a$. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials ${\left(\overline{{\overline{z}}^2+c}\right)}^2+c$. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.

References in corpus (1)

• On (non-)local-connectivity of some Julia sets

Cited by in corpus (6)

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• Schwarz reflections and the Tricorn
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• Invisible Tricorns in Real Slices of Rational Maps