{
  "abstract": "We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II \"Dimension theory of group actions by circle diffeomorphisms II: Minimal sets\". They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup $T$ contained in $\\mathrm{Diff}_+^{1+α}(\\mathbb{S}^1)$ without finite orbits has the same number of minimal sets as its inverse semigroup $T^{-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13254",
  "author_total": 3,
  "authors": [
    {
      "name": "Weikun He"
    },
    {
      "name": "Yuxiang Jiao"
    },
    {
      "name": "Disheng Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aay0jd591ekbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13254",
  "fulltext_url": null,
  "id": "aay0jd591ekbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13254",
  "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": "W7221511004",
  "pdf_url": "https://arxiv.org/pdf/2610.13254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aay0jd591ekbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Dimension theory of group actions by circle diffeomorphisms I: Stationary measures",
  "title_text": "Dimension theory of group actions by circle diffeomorphisms I: Stationary measures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/aay0jd591ekbf",
  "venue": "arXiv",
  "versions": []
}