{
  "abstract": "We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21611",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Vejnar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5mm5zpb2wykj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21611",
  "fulltext_url": null,
  "id": "5mm5zpb2wykj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21611",
  "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"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7213881293",
  "pdf_url": "https://arxiv.org/pdf/2609.21611",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5mm5zpb2wykj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21611",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21611"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The circle as a topological fractal",
  "title_text": "The circle as a topological fractal",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/5mm5zpb2wykj9",
  "venue": "arXiv",
  "versions": []
}