{
  "abstract": "How often does an element of the geometric iterated Galois group of a Lattès map fix a vertex of the tree? Let A/K be an abelian variety over a separably closed field of characteristic 0, G <= Aut(A,0) finite, and alpha an isogeny of degree at least 2 normalising G. The Lattès map on X=A/G has a preimage tree governed by alpha-power torsion; we determine the fixed-point proportion in every dimension. For zeta in G the weight is the reciprocal of the number of fixed points of zeta in alpha-power torsion (zero if infinite); equivalently, it is a product of p-adic absolute values of determinants on the alpha-active subspaces; for scalar alpha=m the proportion is (1/|G|) times the sum over 1 != zeta in G of the product over p dividing m of the p-adic absolute value of det(1 - zeta | H_1(A,Z)). We deduce positivity and vanishing criteria, prove the groups G_infinity are martingale groups, determine the limit law, and reduce groups with translations to the fixed-origin case. In dimension one we determine ten values for scalar multipliers and nontrivial G <= Aut(E,0), and for the eight quadratic Lattès classes the proportions are 1/8, 1/4 and 5/16. Bridy-Jones-Kelsey-Lodge exclude the Lattès case from their main theorem; their Theorem 2.5 periodic-point bound is attained, via reductions modulo split primes p = 1 (mod 4), for the map (x^2+a)/(x^2-a), a^2 = -1, and we settle the exceptional case of the Fariña-Asategui-Radi conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07816",
  "author_total": 1,
  "authors": [
    {
      "name": "Xin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2xczrfet508dk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07816",
  "fulltext_url": null,
  "id": "2xczrfet508dk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07816",
  "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.DS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2xczrfet508dk.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "The fixed-point proportion of Lattès maps",
  "title_text": "The fixed-point proportion of Lattès maps",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/2xczrfet508dk",
  "venue": "arXiv",
  "versions": []
}