{
  "abstract": "We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23972",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiao-Chuan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07t2j3rxfk6a9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23972",
  "fulltext_url": null,
  "id": "07t2j3rxfk6a9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23972",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214054131",
  "pdf_url": "https://arxiv.org/pdf/2609.23972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07t2j3rxfk6a9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Minimal sets for torus homeomorphisms with an irrational circle factor",
  "title_text": "Minimal sets for torus homeomorphisms with an irrational circle factor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/07t2j3rxfk6a9",
  "venue": "arXiv",
  "versions": []
}