{
  "abstract": "We initiate a joining rigidity theory for rational maps on the Riemann sphere $\\mathbb P^1=\\mathbb P^1(\\mathbb C)$. Let $f_1,f_2\\colon\\mathbb P^1\\to\\mathbb P^1$ be rational maps of degree at least $2$, and $μ_1,μ_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,μ_1)$ and $(J(f_2),f_2,μ_2)$, namely ergodic probability measures on $J(f_1)\\times J(f_2)$ which are invariant under $f_1\\times f_2$ and whose marginals are $μ_1$ and $μ_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14890",
  "author_total": 2,
  "authors": [
    {
      "name": "Fabrizio Bianchi"
    },
    {
      "name": "Yan Mary He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0fz462nqfys0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14890",
  "fulltext_url": null,
  "id": "c0fz462nqfys0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14890",
  "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": "W7213332384",
  "pdf_url": "https://arxiv.org/pdf/2609.14890",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0fz462nqfys0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14890",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14890"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Joining rigidity for rational maps",
  "title_text": "Joining rigidity for rational maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/c0fz462nqfys0",
  "venue": "arXiv",
  "versions": []
}