{
  "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 $A^2_1$. The correspondence sends an inner function $I$ to the invariant subspace $[I']$, and in the opposite direction, assigns to a non-zero function $H \\in A^2_1$ 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 $I \\to [I']$ is a bijection.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08809",
  "author_total": 1,
  "authors": [
    {
      "name": "Oleg Ivrii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6t86qy4ajkzxp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08809",
  "fulltext_url": null,
  "id": "6t86qy4ajkzxp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08809",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7212036210",
  "pdf_url": "https://arxiv.org/pdf/2609.08809",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6t86qy4ajkzxp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08809",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08809"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical structures of inner functions II",
  "title_text": "Critical structures of inner functions II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/6t86qy4ajkzxp",
  "venue": "arXiv",
  "versions": []
}