Stable convergence of inner functions

TL;DR

This paper introduces a topology on the set of inner functions with derivatives in the Nevanlinna class, linking the convergence of their critical structures to uniform concentration on Korenblum stars and exploring implications for invariant subspaces.

Contribution

It establishes a new topology based on critical structure convergence and connects it to the behavior of invariant subspaces in weighted Bergman spaces.

Findings

Critical structures converge when uniformly concentrated on Korenblum stars.

Topology governs the behavior of invariant subspaces generated by inner functions.

Uses Liouville's correspondence and builds on Korenblum and Roberts' work.

Abstract

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function.