{
  "abstract": "For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value $\\mathfrak s(X, d, T)$ which quantifies the asymptotic separation of periodic orbits. More precisely, $\\mathfrak s(X,d,T)$ is defined to be the supremum of all $s\\ge 0$ for which there exists a sequence of periodic orbits $\\{\\mathcal O_k\\}_{k=1}^\\infty$ in $(X,d,T)$ such that \\[ \\lim_{k\\to \\infty} \\#\\mathcal O_k = +\\infty \\quad \\text{and} \\quad \\liminf_{k\\to\\infty}\\#\\mathcal O_k\\cdotη(\\mathcal O_k)^s >0, \\] where $η(\\mathcal O_k)$ denotes the smallest distance between distinct points in $\\mathcal O_k$. When the dynamical system is induced by a self-similar iterated function system $\\mathcal F$ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension $s_{\\mathcal F}$ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized $s_{\\mathcal F}$-dimensional Hausdorff measure on the self-similar attractor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16296",
  "author_total": 3,
  "authors": [
    {
      "name": "Derong Kong"
    },
    {
      "name": "Zhiqiang Wang"
    },
    {
      "name": "Daohua Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/63wpsd10eqzd7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16296",
  "fulltext_url": null,
  "id": "63wpsd10eqzd7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16296",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.16296",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/63wpsd10eqzd7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16296",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16296"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Asymptotic separation of periodic orbits and fractal dimension",
  "title_text": "Asymptotic separation of periodic orbits and fractal dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/63wpsd10eqzd7",
  "venue": "arXiv",
  "versions": []
}