{
  "abstract": "We establish a complete characterization of local $C^1$ linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local $C^1$ linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally $C^1$ linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond $C^1$ existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21233",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhicheng Tong"
    },
    {
      "name": "Yong Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af8qbe326szn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21233",
  "fulltext_url": null,
  "id": "af8qbe326szn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21233",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.SP"
    ],
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      "openalex",
      "2610|Mathematical Dynamics and Fractals"
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  "openalex": "W7213865063",
  "pdf_url": "https://arxiv.org/pdf/2609.21233",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af8qbe326szn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21233",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21233"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Complete characterization of planar $C^1$ linearizability and optimal conjugacy regularity",
  "title_text": "Complete characterization of planar C^1 linearizability and optimal conjugacy regularity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/af8qbe326szn3",
  "venue": "arXiv",
  "versions": []
}