On a Family of Rational Perturbations of the Doubling Map

Abstract

Agraïments: The three actors were supported by MTM2011-26995-C02-02 and EU Project MRTN-CT-2006-035651 and FPU AP-2009-4564

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.