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Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity — arXiv:2610.37990

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arXiv (en)

Original

https://arxiv.org/abs/2610.37990

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CC0-1.0

Archived

2026-11-01T02:53:08.690Z

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3658ec95bb410a0b148407d5e65db0df8e27378c3dd433bfc7ac2bee8b948dff

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Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity — arXiv:2610.37990 — arXiv (en), https://arxiv.org/abs/2610.37990 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_445e4a0119ea (Metraipolis archive dc_445e4a0119ea, sha256 3658ec95bb410a0b148407d5e65db0df8e27378c3dd433bfc7ac2bee8b948dff)

@misc{dc445e4a0119ea,
  title = {Symmetry-resolved periods in a cyclic-dominance automaton: separating intrinsic from lattice-induced periodicity — arXiv:2610.37990},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.37990},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_445e4a0119ea; sha256 3658ec95bb410a0b148407d5e65db0df8e27378c3dd433bfc7ac2bee8b948dff},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.37990. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: fisica e astronomia (elenco arXiv cond-mat.stat-mech, nuovi lavori del giorno).
Categorie arXiv: Dynamical Systems (math.DS) ; Statistical Mechanics (cond-mat.stat-mech); Pattern Formation and Solitons (nlin.PS).
Pagina del lavoro: https://arxiv.org/abs/2610.37990 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:19:58.605Z dalla pagina https://arxiv.org/list/cond-mat.stat-mech/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).