Critical values of inner functions

TL;DR

This paper investigates the boundary behavior of inner functions with finite entropy, establishing the existence of radial limits at almost every boundary point and demonstrating the continuous variation of a related singular measure.

Contribution

It proves the existence of radial and minimal fine limits for finite entropy inner functions and shows the continuous dependence of the singular value measure on the function.

Findings

Inner functions have radial limits at almost every boundary point.

The singular value measure varies continuously with the inner function.

A connection between Beurling-Carleson sets and angular derivatives is established.

Abstract

Let $\mathscr J$ be the space of inner functions of finite entropy endowed with the topology of stable convergence. We prove that an inner function $F \in \mathscr J$ possesses a radial limit (and in fact, a minimal fine limit) in the unit disk at $\sigma(F')$ a.e. point on the unit circle. We use this to show that the singular value measure $\nu(F) = \sum_{c \in \text{crit } F} (1-|c|) \cdot \delta_{F(c)} + F_*(\sigma(F'))$ varies continuously in $F$. Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.