Critical structures of inner functions

TL;DR

This paper explores the parametrization of inner functions, extending classical finite Blaschke product results to infinite degree cases using invariant subspaces and PDE techniques.

Contribution

It introduces a novel approach to parametrize inner functions via invariant subspaces, linking complex analysis with nonlinear elliptic PDEs.

Findings

Extension of finite Blaschke product characterization to infinite degree

Parametrization of inner functions using 1-generated invariant subspaces

Application of Liouville correspondence to connect complex analysis and PDEs

Abstract

A celebrated theorem of M. Heins says that up to post-composition with a M\"obius transformation, a finite Blaschke product is uniquely determined by its critical points. K. Dyakonov suggested that it may interesting to extend this result to infinite degree, however, one needs to be careful since inner functions may have identical critical sets. In this work, we try parametrizing inner functions by 1-generated invariant subspaces of the weighted Bergman space $A^2_1$. Our technique is based on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.