{
  "id": "1903.03551",
  "version": "v1",
  "published": "2019-03-08T16:52:25.000Z",
  "updated": "2019-03-08T16:52:25.000Z",
  "title": "Generalized fractal dimensions of invariant measures of full-shift systems over uncountable alphabets: generic behavior",
  "authors": [
    "Silas Luiz Carvalho",
    "Alexander Condori"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each $q>0$, zero lower $q$-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system $(X,T)$ (where $X= M^{\\Z}$ is endowed with a sub-exponential metric and the alphabet $M$ is a perfect and compact metric space), for which we show that a typical invariant measure has, for each $q>1$, infinite upper $q$-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each $s\\in(0,1)$ and each $q>1$, zero lower $s$-generalized and infinite upper $q$-generalized dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-08T16:52:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28D05",
      "37B20",
      "37B10"
    ],
    "keywords": [
      "generalized fractal dimension",
      "full-shift system",
      "typical invariant measure",
      "generic behavior",
      "uncountable alphabets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}