{
  "id": "2410.23770",
  "version": "v1",
  "published": "2024-10-31T09:34:19.000Z",
  "updated": "2024-10-31T09:34:19.000Z",
  "title": "Cellular automata, percolation and dynamical dichotomies",
  "authors": [
    "Sebastián Barbieri",
    "Felipe García-Ramos",
    "Siamak Taati"
  ],
  "comment": "23 pages, 1 cute picture",
  "categories": [
    "math.DS",
    "math.GR",
    "math.PR"
  ],
  "abstract": "We establish a connection between percolation on the Cayley graphs of a group and the dynamical diversity of cellular automata on that group. Specifically, we demonstrate that Gilman's dichotomy between equicontinuity and sensitivity with respect to Bernoulli measures holds on a finitely generated group if and only if the group has a trivial percolation threshold. Consequently, we show that a countable group satisfies Gilman's dichotomy if and only if it is locally virtually cyclic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-31T09:34:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B15",
      "37A50",
      "60K35"
    ],
    "keywords": [
      "cellular automata",
      "dynamical dichotomies",
      "countable group satisfies gilmans dichotomy",
      "trivial percolation threshold",
      "bernoulli measures holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}