{
  "id": "1512.01982",
  "version": "v1",
  "published": "2015-12-07T11:14:52.000Z",
  "updated": "2015-12-07T11:14:52.000Z",
  "title": "Topological Entropy on Points without Physical-like Behaviour",
  "authors": [
    "Eleonora Catsigeras",
    "Xueting Tian",
    "Edson Vargas"
  ],
  "comment": "11 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $f:M\\rightarrow M$ be a $C^1$ diffeomorphism on a compact Riemannian manifold $M$. Let $\\mathcal O_f$ denote the space of all SRB-like measures and for $x\\in M$, $pw(x)$ denote the limit set of $ \\{\\frac1n\\sum_{j=0}^{n-1}\\delta_{f^j(x)} \\}_{n\\in\\mathbb{N}} $ in weak$^*$ topology where $\\delta_y$ is the Dirac probability measure supported at $y\\in M.$ We state a sufficient condition to prove that the set of points without physical-like behaviour $$\\Gamma_f=\\{x: pw(x)\\cap \\mathcal{O}_{f}=\\emptyset\\}$$ has full topological entropy, even though in general it always has zero Lebesgue measure. In particular, this phenomena is valid for all $C^1$ transitive Anosov diffeomorphisms and time$-1$ maps of all $C^1$ transitive Anosov flows. We emphasize that the system is just required $C^1.$ The proof ideas are mainly based on Pesin's entropy formula and variational principle of saturated sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-07T11:14:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D20",
      "37D30",
      "37C45",
      "37A35",
      "37B40"
    ],
    "keywords": [
      "physical-like behaviour",
      "dirac probability measure",
      "compact riemannian manifold",
      "zero lebesgue measure",
      "pesins entropy formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}