{
  "abstract": "We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\\mathbb{R}^n$, $n\\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of $\\overline{\\mathbb{R}^n}$ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00312",
  "author_total": 1,
  "authors": [
    {
      "name": "Alastair N. Fletcher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/antzv5txsnwzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00312",
  "fulltext_url": null,
  "id": "antzv5txsnwzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00312",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7216228793",
  "pdf_url": "https://arxiv.org/pdf/2610.00312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/antzv5txsnwzv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps",
  "title_text": "Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/antzv5txsnwzv",
  "venue": "arXiv",
  "versions": []
}