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 . 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 . In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.