{
  "abstract": "Let $F(x,y)=(f(x),g(x,y))$ be a continuous triangular map of the unit square and let $z=(x_0,y_0)$ be a point such that $ω_f(x_0)$ is a periodic orbit $\\{p_0,\\dots,p_{m-1}\\}$ of $f$. Then $ω_F(z)=\\bigcup_{i}\\{p_i\\}\\times M_i$ for nonempty compact sets $M_i$, and we ask which $m$-tuples $(M_0,\\dots,M_{m-1})$ occur. For $m=1$ the answer is a theorem of Kolyada and Snoha. We show that a tuple occurs if and only if there are continuous surjections $M_i\\to M_{i+1}$, indices modulo $m$, whose composition around the cycle is chain transitive. This is a cyclic version of classical facts due to Dowker and Friedlander, Sharkovsky, Hirsch, Smith and Zhao, and Kolyada and Snoha. It yields two necessary conditions: no $M_i$ is one of the exceptional sets of Kolyada and Snoha, and there are continuous surjections $M_0\\to M_1\\to\\dots\\to M_{m-1}\\to M_0$. We prove that these conditions are also sufficient when every $M_i$ has uncountably many connected components, and when every $M_i$ has countably many components of which only finitely many are nondegenerate. In particular the sections of one $ω$-limit set need not be homeomorphic. Every tuple satisfying the two conditions falls into one of these cases or into a single remaining class, in which the sections have countably many components and infinitely many of them are nondegenerate; that class is studied in a companion paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06889",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Luis García Guirao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/10s3mzhe5xyq9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06889",
  "fulltext_url": null,
  "id": "10s3mzhe5xyq9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06889",
  "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/2611.06889",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/10s3mzhe5xyq9.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06889",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06889"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "$ω$-limit sets of triangular maps over periodic orbits of the base",
  "title_text": "ω-limit sets of triangular maps over periodic orbits of the base",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:31Z",
  "url": "https://research.draconitic.com/paper/10s3mzhe5xyq9",
  "venue": "arXiv",
  "versions": []
}