{
  "abstract": "Let $E$ be the Cantor set, $G=\\operatorname{Homeo}(E)$, let $P\\le G$ be the group of order preserving homeomorphisms and $L\\le G$ the group of homeomorphisms preserving the Cantor--Lebesgue measure $λ$. We study how $G$ is built from these two subgroups. In 2003, Glasner and Weiss stated that $G=LP=PL$. We show that this is not the case, in fact $PL$ and $LP$ are two different proper subsets of $G$. On the other hand, $PL$ is a dense $G_δ$ subset of $G$, the multiplication map $P\\times L\\to PL$ is a homeomorphism, and $G=PLP=LPL$. In particular, every homeomorphism of $E$ is conjugate to an element of $PL$. We classify the double cosets $PgL$ using special clopen values set. There are continuum many of them, and $PL$ is the only one which is not meager. We prove that $G$ is topologically generated by one element of $P$ and one element of $L$ and these two generators produce the Thompson's group $V$. We also show that the generic homeomorphism of $E$ is the product of a generic element of $P$ and a generic element of $L$. The equality $G=PL$ fails also if $λ$ is replaced by any other full nonatomic measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32424",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Vejnar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17262hdwr1hhd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32424",
  "fulltext_url": null,
  "id": "17262hdwr1hhd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32424",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223680132",
  "pdf_url": "https://arxiv.org/pdf/2610.32424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17262hdwr1hhd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On the homeomorphism group of the Cantor set: monotone and measure preserving maps",
  "title_text": "On the homeomorphism group of the Cantor set: monotone and measure preserving maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/17262hdwr1hhd",
  "venue": "arXiv",
  "versions": []
}