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Critical structures of inner functions II

  • Oleg Ivrii
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Abstract

We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space A21. The correspondence sends an inner function I to the invariant subspace [I'], and in the opposite direction, assigns to a non-zero function H ∈ A21 the Liouville map IH associated to the canonical solution of the Gauss curvature equation Δu = |H|2 e2u. We prove that I'H generates the same cyclic subspace as H. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence I → [I'] is a bijection.

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Cite

BibTeXRIS
BibTeX
@misc{ivrii2026critical,
  title = {{Critical structures of inner functions II}},
  author = {Oleg Ivrii},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08809},
  eprint = {2609.08809},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08809},
}
RIS
TY  - UNPB
TI  - Critical structures of inner functions II
AU  - Oleg Ivrii
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08809
UR  - https://arxiv.org/abs/2609.08809
AB  - We study the correspondence proposed in \cite{critical-structures} between inner functions modulo post-compositions with automorphisms of the unit disk and cyclic subspaces of the weighted Bergman space A^2_1. The correspondence sends an inner function I to the invariant subspace [I'], and in the opposite direction, assigns to a non-zero function H ∈ A^2_1 the Liouville map I_H associated to the canonical solution of the Gauss curvature equation Δu = |H|^2 e^2u. We prove that I'_H generates the same cyclic subspace as H. Combined with the results in \cite{critical-structures}, this shows that these two mappings are inverses of one another, and hence the correspondence I → [I'] is a bijection.
ER  -