{
  "id": "2401.08440",
  "version": "v2",
  "published": "2024-01-16T15:37:42.000Z",
  "updated": "2024-09-17T07:22:12.000Z",
  "title": "Local mean dimension theory for sofic group actions",
  "authors": [
    "Felipe García-Ramos",
    "Yonatan Gutman"
  ],
  "comment": "to appear in GGD",
  "categories": [
    "math.DS"
  ],
  "abstract": "Using a local perspective, we introduce \\textit{mean dimension pairs} and give sufficient conditions of when every non-trivial factor of a continuous group action of a sofic group $G$ has positive mean dimension. In addition we show that the mean dimension map is Borel, and that the set of subshifts with completely positive mean dimension of $[0,1]^G$, the full $G$-shift on the interval, is a complete coanalytic set in the set of all subshifts (hence not Borel). Our results are new even when the acting group is $\\Z$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-17T07:22:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37B40",
      "54H05"
    ],
    "keywords": [
      "local mean dimension theory",
      "sofic group actions",
      "positive mean dimension",
      "mean dimension map",
      "complete coanalytic set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}