{
  "abstract": "We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $β=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $ν_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\\in\\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_ϕ(n\\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\\tfrac12(δ_{-1}+δ_1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24828",
  "author_total": 1,
  "authors": [
    {
      "name": "Farrukh Mukhamedov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fnmc4vcw7mb5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24828",
  "fulltext_url": null,
  "id": "3fnmc4vcw7mb5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24828",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214036213",
  "pdf_url": "https://arxiv.org/pdf/2609.24828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fnmc4vcw7mb5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Sparse empirical measures for non-statistical interval maps with two neutral fixed points",
  "title_text": "Sparse empirical measures for non-statistical interval maps with two neutral fixed points",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/3fnmc4vcw7mb5",
  "venue": "arXiv",
  "versions": []
}