  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.
