{
  "id": "1908.00998",
  "version": "v1",
  "published": "2019-08-02T18:21:22.000Z",
  "updated": "2019-08-02T18:21:22.000Z",
  "title": "A note on the fractal dimensions of invariant measures associated with $C^{1+α}-diffeomorphisms, expanding and expansive homeomorphisms",
  "authors": [
    "Alexander Condori",
    "Silas L. Carvalho"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show in this work that the upper and the lower generalized fractal dimensions $D^{\\pm}_{\\mu}(q)$, for each $q\\in\\mathbb{R}$, of an ergodic measure associated with an invertible bi-Lipschitz transformation over a Polish metric space are equal, respectively, to its packing and Hausdorff dimensions. This is particularly true for hyperbolic ergodic measures associated with $C^{1+\\alpha}$-diffeomorphisms of smooth compact Riemannian manifolds, from which follows an extension of Young's Theorem (Young, L., S. Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems, 2(1):109-124, 1982). Analogous results are obtained for expanding systems. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures with zero correlation dimension, under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\\ge 0$, $D^{+}_{\\mu}(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-02T18:21:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28D05",
      "37B20",
      "37B10",
      "37C05"
    ],
    "keywords": [
      "invariant measures",
      "expansive homeomorphisms",
      "smooth compact riemannian manifolds",
      "hyperbolic metric",
      "zero correlation dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}