On a Family of Degree 4 Blaschke Products [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Jordi Canela


We investigate the parameter plane of the rational family of perturbations of the doubling map given by the Blaschke products $B_a(z)=z^3\frac{z-a}{1-\bar{a}z}$. We study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter $a$. We also classify and study the different hyperbolic components of the parameter plane according to the critical orbits. The non holomorphic dependence on the parameters of the family gives rise to the existence of phenomena which cannot occur otherwise. For $|a|\geq 2$, there appear tongues as sets of parameters with attracting cycles in the circle. We study the connectivity of these tongues and how bifurcations take place along their boundaries. Also for parameters with $|a|>2$, small ``copies'' of the connectedness locus of the antipolynomials $p_c(z)=\overline{z}^2+c$, the Tricorn, appear contained in regions of parameters for which the free critical points enter and exit the unit disk. We use techniques of quasiconformal surgery to explore the relation between such $B_a$ and the degree 4 polynomials $p_c^2(z)=\left(\overline{\overline{z}^2+c}\right)^2+c$.