{
  "abstract": "In this article, we study algebraic relations among rational maps on $\\mathbb{P}^1$ and among their preperiodic points. Working over $\\mathbb{C}$, we characterize algebraic families of positive-dimensional varieties in $(\\mathbb{P}^1)^N$ and tuples of maps $(f_1, \\ldots, f_N)$ for which the subvariety is dynamically special for a Zariski-dense set of parameters. Moreover, outside of well-understood elliptic phenomena, we show that all families of maps and subvarieties that bifurcate are maximally bifurcating. Working over $\\overline{\\mathbb{Q}}$, we characterize arithmetic and geometric potential bigness for the canonical line bundle associated to families of endomorphisms of $(\\mathbb{P}^1)^N$ when restricted to the subvarieties. As consequences, we obtain uniform bounds for the periods and preperiods of relations among non-exceptional maps, and we show uniform bounds on the geometry of preperiodic points satisfying these relations. Over $\\overline{\\mathbb{Q}}$, our results yield corresponding uniform bounds for points and subvarieties satisfying a height gap.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40147",
  "author_total": 2,
  "authors": [
    {
      "name": "Laura DeMarco"
    },
    {
      "name": "Niki Myrto Mavraki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3s2hxkz0d96nw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40147",
  "fulltext_url": null,
  "id": "3s2hxkz0d96nw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40147",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40147",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3s2hxkz0d96nw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40147",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40147"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Dynamical relations on $\\mathbb{P}^1$, higher bifurcations, and arithmetic bigness",
  "title_text": "Dynamical relations on ℙ^1, higher bifurcations, and arithmetic bigness",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/3s2hxkz0d96nw",
  "venue": "arXiv",
  "versions": []
}