{
  "abstract": "In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of $\\mathsf{PO}(2, 1) \\times \\mathsf{PO}(2, 1)$, the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of $\\mathsf{PO}(2, 1) \\times \\mathsf{PO}(2, 1)$, its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $λ$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $λ$. We hope this will be a starting point for a more general study of extremal entropy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10325",
  "author_total": 2,
  "authors": [
    {
      "name": "Richard Canary"
    },
    {
      "name": "Dongryul Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5x7vk06r9wdb0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10325",
  "fulltext_url": null,
  "id": "5x7vk06r9wdb0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10325",
  "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.GT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7212117753",
  "pdf_url": "https://arxiv.org/pdf/2609.10325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5x7vk06r9wdb0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Extremal entropy for products of Fuchsian representations",
  "title_text": "Extremal entropy for products of Fuchsian representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/5x7vk06r9wdb0",
  "venue": "arXiv",
  "versions": []
}