Stable convergence of inner functions

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Abstract

Let (Formula presented.) be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on (Formula presented.) where (Formula presented.) if the critical structures of (Formula presented.) converge to the critical structure of (Formula presented.). We show that this occurs precisely when the critical structures of the (Formula presented.) 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.

Original languageEnglish
Pages (from-to)257-286
Number of pages30
JournalJournal of the London Mathematical Society
Volume102
Issue number1
DOIs
StatePublished - 1 Aug 2020

Keywords

  • 30C80
  • 30F45 (primary)
  • 30H20 (secondary)
  • 30J05

All Science Journal Classification (ASJC) codes

  • General Mathematics

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