Dynamics and bifurcations of a Family of Rational Maps with parabolic Fixed Points

We study a family of rational maps of the sphere with the property that each map has two fixed points with multiplier −1; moreover each map has no period 2 orbits. The family we analyze is Ra(z) = z3−z −z2+az+1 , where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).