{
  "id": "2309.07575",
  "version": "v1",
  "published": "2023-09-14T10:11:23.000Z",
  "updated": "2023-09-14T10:11:23.000Z",
  "title": "On the generalized dimensions of chaotic attractors",
  "authors": [
    "Théophile Caby",
    "Michele Gianfelice"
  ],
  "comment": "34 pages, 6 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove that if $\\mu$ is the physical measure of a chaotic flow diffeomorphically conjugated to a suspension flow based on a Poincar\\'{e} application $R$ with physical measure $\\mu_{R}$, $D_{q}(\\mu)=D_{q}(\\mu _{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q$ with $q \\neq1$. We also prove that a similar result holds for the local dimensions so that, under the additional hypothesis of exact-dimensionality of $\\mu_{R}$, our result extends to the case $q=1$. We then apply these results to estimate the $D_{q}$ spectrum of the R\\\"ossler attractor and prove the existence of the information dimension $D_{1}$ for the Lorenz '63 flow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-14T10:11:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37D45",
      "37E05"
    ],
    "keywords": [
      "generalized dimension",
      "chaotic attractors",
      "physical measure",
      "similar result holds",
      "result extends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}