Möbius transformations and Blaschke products: The geometric connection

Let B be a degree-n Blaschke product and, for λ∈T, let z1,λ,…,zn,λ, ordered according to increasing argument, denote the (distinct) solutions to B(z)−λ=0. Then there is a smooth curve C such that for each λ the line segments joining zj,λ and zj+1,λ are tangent to C. We study the situation in which C is an ellipse and describe the relation to the action of the points zj,λ 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.