{
  "abstract": "For $Q_α(z)=e^{2πiα}z+z^2$ with $α$ irrational, we prove that the analytic centraliser of $Q_α$ is trivial whenever $Q_α$ is not analytically linearizable. Together with the quadratic Brjuno--Yoccoz theorem this gives a complete dichotomy, for the centraliser is infinite cyclic at the non-Brjuno parameters and conjugate to the group of nonzero complex dilations at the Brjuno parameters. The result extends the dense-parameter conclusion of Avila--Cheraghi--Eliad to every nonlinearizable member of the quadratic family. The proof starts from Inou's graph-globalization construction, where we retain the original neutral fixed point on a Riemann surface and verify the branch topology and finite-valence estimate, including multiplicities. Nonlinearizability, through the equality case of Schwarz's lemma, forces the surface to be the plane, and a direct quadratic growth estimate then makes the two entire projections polynomial, so that the established polynomial commutant classification yields the integral-iterate conclusion. We also determine the convergence set of the formal commutants and classify the analytic iterative roots of every nonzero iterate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18667",
  "author_total": 1,
  "authors": [
    {
      "name": "Wen Si"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fp26pkes9a05e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18667",
  "fulltext_url": null,
  "id": "fp26pkes9a05e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18667",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221769592",
  "pdf_url": "https://arxiv.org/pdf/2610.18667",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fp26pkes9a05e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18667",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18667"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Analytic centralisers of quadratic germs and the Brjuno dichotomy",
  "title_text": "Analytic centralisers of quadratic germs and the Brjuno dichotomy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/fp26pkes9a05e",
  "venue": "arXiv",
  "versions": []
}