{
  "abstract": "We show that the critical cusp exponent $β=-1/2$ admits $W^{2,1}$ invariant densities in an explicit family of expanding full-branch interval maps. The usual estimates exclude this exponent because two zero-order coefficients in the second derivative of the transfer operator are individually non-integrable. Their signed combination, however, contains the factor $β(2β+1)$ for a pure power cusp. We exploit its cancellation at $β=-1/2$ to prove a $W^{2,1}$ Lasota--Yorke inequality and a spectral gap for the explicit family. We also identify a separate obstruction caused by an interior cusp value. As an application, a small Bernoulli probability perturbation between two fixed maps has a stationary density that depends real-analytically on the perturbation parameter in $W^{2,1}$, with explicit formulas for all response derivatives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Acta NUUz (Exact Sciences), 2026, 2/3, 178-183",
  "arxiv": "2610.23180",
  "author_total": 2,
  "authors": [
    {
      "name": "Davrbek Oltiboev"
    },
    {
      "name": "Furqat Vafoyev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ntn4nqxmaffh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23180",
  "fulltext_url": null,
  "id": "7ntn4nqxmaffh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23180",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222975558",
  "pdf_url": "https://arxiv.org/pdf/2610.23180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ntn4nqxmaffh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Critical cusp cancellation and analytic response for a Bernoulli random interval map",
  "title_text": "Critical cusp cancellation and analytic response for a Bernoulli random interval map",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/7ntn4nqxmaffh",
  "venue": "arXiv",
  "versions": []
}