Moebius transformations and Blaschke products: the geometric connection.
Let B be a degree-n Blaschke product and, for a complex number l of modulus 1, let z1l, ... znl ordered according to increasing argument, denote the (distinct) solutions to B(z) - l = 0. Then there is a smooth curve C such that for each l the line segments joining zjl and z(j+1)l are tangent to C. We study the situation in which C is an ellipse and describe the relation to the action of the points zjl under elliptic disk automorphisms. These results provide a condition for the numerical range of a compressed shift operator with finite Blaschke symbol to be an elliptical disk. We also consider infinite Blaschke products and the action of parabolic and hyperbolic disk automorphisms
Linear Algebra Appl.
Gorkin, Pamela; Daepp, Ulrich; Shaffer, Andrew; and Voss, Karl. "Moebius transformations and Blaschke products: the geometric connection.." Linear Algebra Appl. (2017) : 186-211.