{
  "id": "2012.15372",
  "version": "v1",
  "published": "2020-12-30T23:56:29.000Z",
  "updated": "2020-12-30T23:56:29.000Z",
  "title": "$G$-index, topological dynamics and marker property",
  "authors": [
    "Masaki Tsukamoto",
    "Mitsunobu Tsutaya",
    "Masahiko Yoshinaga"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DS",
    "math.AT"
  ],
  "abstract": "Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical system, we study the $\\mathbb{Z}_p$-index of the set of $p$-periodic points. We find that its growth is at most linear in $p$. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-30T23:56:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "55M35"
    ],
    "keywords": [
      "marker property",
      "topological dynamics",
      "fixed-point free dynamical system",
      "index theory",
      "index roughly measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}