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 z _1l , ... z _nl 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 z _jl 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 z _jl 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