{
  "abstract": "We prove a uniform form of the dynamical Manin-Mumford conjecture for split endomorphisms $Φ= \\left(f_1,\\ldots,f_n\\right): \\left(\\mathbb{P}^1\\right)^n \\to \\left(\\mathbb{P}^1\\right)^n$ over $\\mathbb{C}$, where each $f_i$ has degree $d \\geq 2$ and at most one $f_i$ is exceptional. Specifically, we bound the number of irreducible components of $\\overline{\\operatorname{Prep}\\left(Φ\\right)\\cap V\\left(\\mathbb{C}\\right)}^{\\operatorname{Zar}}$ uniformly, depending only on $\\left(n,d,\\operatorname{deg} V\\right)$. We also obtain uniform statements for the dynamical Bogomolov conjecture. Our approach combines a degeneration argument via ultrafilters with a new non-archimedean rigidity theorem, along with the quantitative dynamical Zhang's inequality by Mavraki and the second author.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40152",
  "author_total": 2,
  "authors": [
    {
      "name": "Jit Wu Yap"
    },
    {
      "name": "Zhuchao Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25kty0pda2ts5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40152",
  "fulltext_url": null,
  "id": "25kty0pda2ts5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40152",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40152",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25kty0pda2ts5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40152",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40152"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for $\\left(\\mathbb{P}^1\\right)^n$",
  "title_text": "Non-Archimedean Rigidity and the Uniform Dynamical Manin--Mumford for (ℙ^1)^n",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/25kty0pda2ts5",
  "venue": "arXiv",
  "versions": []
}