Unicritical Blaschke products and domains of ellipticity (Q2342874)

The aim of this paper is to study and classify Blaschke products with a unique critical point in the unit disk. In the same way as for Möbius transforms, there is a classification of finite Blaschke products that characterizes their corresponding Julia sets as elliptic, hyperbolic or parabolic. After conjugation and reduction to a simpler form depending on the parameters, the author studies the domain of ellipticity and the connectedness locus of these Blaschke products, in terms of such parameters. It is proved that the connectedness locus plays the role of the Mandelbrot set for the family of Blaschke products. In addition, it is shown how the connectedness locus can be obtained by adding one point to the domain of ellipticity.