Describing Blaschke Products by Their Critical Points

In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. Dyakonov suggested that it may be interesting to extend this result to infinite degree, however, one must be careful since infinite Blaschke products may have identical critical sets. I will first explain how to parametrize inner functions F of finite entropy (i.e. which have derivative in the Nevanlinna class) in terms of InnF′. The answer involves measures on the unit circle that do not charge Beurling–Carleson sets. Afterwards, I will discuss how one might parametrize arbitrary inner functions using 1-generated invariant subspaces of the weighted Bergman space A12. The proofs rely on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.