Stable convergence of inner functions
| Autor: | Oleg Ivrii |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Class (set theory) Mathematics - Complex Variables Mathematics::Complex Variables General Mathematics 010102 general mathematics 30J05 30F45 30C80 30H20 Function (mathematics) Natural topology 01 natural sciences Linear subspace Bergman space 0103 physical sciences Convergence (routing) FOS: Mathematics 010307 mathematical physics Complex Variables (math.CV) 0101 mathematics Invariant (mathematics) Topology (chemistry) Mathematics |
| Zdroj: | Journal of the London Mathematical Society. 102:257-286 |
| ISSN: | 1469-7750 0024-6107 |
| DOI: | 10.1112/jlms.12319 |
| Popis: | 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. 45 pages |
| Databáze: | OpenAIRE |
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