{
  "annotation_id": "5a1039a5-bf3a-48dd-b31e-99c64f947e6c",
  "date_created": "2026-02-17T05:53:19.896000Z",
  "date_modified": "2026-02-17T05:53:19.896000Z",
  "file_hash": "03cde5fd1bcb8b50c7e2e4023f47ea90e3fbae0437c5bfc99fb43d8753abfe75",
  "private": false,
  "record": {
    "abstract": "The Emergent Order Spectrum $\\Omega(x,y)$ is a topological invariant of dynamical systems providing order-types induced by the limit order of order-compatible nested $\\varepsilon_n$-chains (with $\\varepsilon_n\\to 0$) from $x$ to $y$. In this paper, we investigate how rich these spectra can be under natural dynamical hypotheses. For a transitive homeomorphism $f$ on a compact metric space $X$ with $\\lvert X\\rvert=\\mathfrak{c}$, we show that the global spectrum $\\Omega_f(X^2)$ is universal at the countable scattered level: every countable scattered order-type together with the order-type of the rationals appear in $\\Omega_f(X^2)$. More precisely, there exists a comeagre subset $M\\subseteq X^2$ such that, for every $(x,y)\\in M$, the individual spectrum $\\Omega_f(x,y)$ already realizes all countably infinite scattered order-types; moreover, the order-type of the rationals belongs to $\\Omega_f(x,y)$ for every pair $(x,y)\\in X^2$.",
    "arxiv_id": "2601.09325",
    "authors": [
      "Filippo Ciavattini",
      "Marco Farotti",
      "Camilla Lucamarini"
    ],
    "categories": [
      "math.DS"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Emergent order spectrum for transitive homeomorphisms",
    "url": "https://arxiv.org/abs/2601.09325",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "cce9b191-c009-4243-a72d-4019517e0630",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}