{
  "abstract": "Let $μ$ be a finitely supported probability measure on $\\mathrm{SL}_2(\\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\\mathbb{P}^1(\\mathbb{R})$ is singular with respect to the $\\mathrm{SO}(2)$-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17506",
  "author_total": 1,
  "authors": [
    {
      "name": "Timothée Bénard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cb59jt6mk18hn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17506",
  "fulltext_url": null,
  "id": "cb59jt6mk18hn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17506",
  "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",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7213432540",
  "pdf_url": "https://arxiv.org/pdf/2609.17506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cb59jt6mk18hn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Proof of the singularity conjecture for discrete subgroups of $\\mathrm{SL}_2(\\mathbb{R})$",
  "title_text": "Proof of the singularity conjecture for discrete subgroups of SL_2(ℝ)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/cb59jt6mk18hn",
  "venue": "arXiv",
  "versions": []
}