Critical values of inner functions

2022/12/30 by Ivrii, Oleg, Kreitner, Uri
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.14818

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 ∈ \mathscr J possesses a radial limit (and in fact, a minimal fine limit) in the unit disk at σ(F') a.e. point on the unit circle. We use this to show that the singular value measure ν(F) = ∑_{c ∈ crit F} (1-|c|) ⋅ δ_F(c) + F_*(σ(F')) varies continuously in F. Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.

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