{
  "abstract": "Periodic attractors of spatially extended systems possess not one period but a hierarchy of periods, one for each level of observation and symmetry quotient, and this hierarchy separates how a pattern evolves from how it fits the finite domain. For a synchronous threshold model of cyclic dominance on an $L\\times L$ torus, equivalent to the three-color cyclic cellular automaton of Fisch, Gravner and Griffeath, we measure for every attractor a \\emph{tuple} of periods: that of the populations ($\\Tpop$), of the full configuration ($\\Tst$), and of the configuration up to color rotations ($\\Tcol$), lattice symmetries ($\\Tgeo$) and the full space--color group ($\\TG$). Equivariance implies exact divisibility relations and the factorization $\\Tst=\\TG\\,m$, where $m$ counts the applications of the symmetry $h$ that maps the configuration onto itself after $\\TG$ steps: $\\TG$ and the type of $h$ belong to the pattern, $m$ to the lattice, and the attractors are relative periodic orbits whose full period emerges from the interaction between the intrinsic dynamics and the lattice symmetries. A family of attractors built around a pair of counter-rotating spirals, with $\\Tpop=24$ and $\\TG=8$, persists from $L=9$ to $L=40$ while $\\Tst$ takes values from 24 to 960, given in every case by the order of $h$; on a rectangular $9\\times12$ torus it takes the value 48, absent from all square tori. With more than $4.5\\times10^5$ runs at each of $L=8$ and $L=9$ we estimate the basins of attraction: fixed points take 95--97\\% of the configuration space, and no single period is enough to tell the oscillating attractors apart. The same analysis locates local diversity: it is balanced whenever the cycle has a color symmetry and, on large lattices, it is concentrated at the spiral cores.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37990",
  "author_total": 2,
  "authors": [
    {
      "name": "Francisco J. Muñoz"
    },
    {
      "name": "Juan Carlos Nuño"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6y859jhhqw3qf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37990",
  "fulltext_url": null,
  "id": "6y859jhhqw3qf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37990",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "nlin.PS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37990",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6y859jhhqw3qf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.37990",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37990"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity",
  "title_text": "Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6y859jhhqw3qf",
  "venue": "arXiv",
  "versions": []
}