{
  "abstract": "Let $F(x,y)=(f_2(y),f_1(x))$ be a Cournot map of the unit square, $φ=f_2\\circ f_1$ and $χ=f_1\\circ f_2$. Linero Bas and Muñoz Guillermo proved that an $ω$-limit set of $F$ with nonempty interior is connected if and only if \\textup{(C1)} its two projections are connected and \\textup{(C2)} $ω_{F^2}(z)\\capω_{F^2}(F(z))\\neq\\emptyset$, and conjectured that this holds for every $ω$-limit set. We prove the conjecture for every point $z=(x,y)$ such that $ω_φ(x)$ or $ω_χ(y)$ is finite, and for Lebesgue almost every point $z$ of the square whenever $φ$ is metrically regular (every metric attractor is a periodic orbit, a solenoid, or a cycle of intervals carrying a well-behaved absolutely continuous invariant measure). Over a periodic orbit the $ω$-limit set is a \\emph{comb}, whose sections are rigid; when one projection is connected it is a point, a \\emph{lattice} $(P\\times J)\\cup(K\\times Q)$ with $P\\subset K$, $Q\\subset J$ and $\\#P=\\#Q$, or an \\emph{alternating comb} of $2m$ disjoint segments. Conversely, every lattice and every alternating comb is an $ω$-limit set of a Cournot map with piecewise linear reaction functions; the construction rests on a lemma producing exact compositions of piecewise linear maps with prescribed values. This answers, for this class, the question of Balibrea, Linero and Cánovas of which nowhere dense sets are $ω$-limit sets of Cournot maps. For instance, the letter H is not an $ω$-limit set at any such point, and for metrically regular $φ$ the points whose $ω$-limit set is connected, infinite, nowhere dense and not a lattice form a Lebesgue null set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27229",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Luis García Guirao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bbbd0bdtf5x3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27229",
  "fulltext_url": null,
  "id": "0bbbd0bdtf5x3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27229",
  "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"
    ],
    [
      "arxiv",
      "nlin.PS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223137640",
  "pdf_url": "https://arxiv.org/pdf/2610.27229",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bbbd0bdtf5x3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.27229",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27229"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.PS",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Connected $ω$-limit sets of Cournot maps: periodic orbits, realization and almost every point",
  "title_text": "Connected ω-limit sets of Cournot maps: periodic orbits, realization and almost every point",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0bbbd0bdtf5x3",
  "venue": "arXiv",
  "versions": []
}