Critical values of inner functions.(English)Zbl 07916603
Summary: 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 \(\sigma (F^\prime)\) a.e. point on the unit circle. We use this to show that the singular value measure \(\nu(F) = \sum_{c \in \operatorname{crit} F}(1 - |c|) \cdot \delta_{F (c)} + F_\ast(\sigma (F^\prime))\) varies continuously in \(F\). Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.
MSC:
| 30Dxx | Entire and meromorphic functions of one complex variable, and related topics |
| 30Jxx | Function theory on the disc |
| 30Cxx | Geometric function theory |
Cite
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