{
  "id": "2112.11149",
  "version": "v2",
  "published": "2021-12-21T12:37:43.000Z",
  "updated": "2023-09-20T21:27:40.000Z",
  "title": "On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems",
  "authors": [
    "Reza Mohammadpour"
  ],
  "comment": "Minor mistakes were corrected, the title was changed, and the article was reorganized. The main result was proven under the assumption that the cocycle $Df_{|E^{cu}(f)}^{-1}$ had a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure",
  "categories": [
    "math.DS",
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds of arbitrary dimension that admit a dominated splitting $E^{cs} \\oplus E^{cu}.$ We prove that if the Lyapunov exponents along $E^{cu}$ are positive for Lebesgue almost every point, then a map $f$ is non-uniformly expanding along $E^{cu}$ under the assumption that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. As a result, there exists a physical SRB measure for a $C^{1+\\alpha}$ diffeomorphism map $f$ that admits a dominated splitting $E^{cs} \\oplus E^{cu}$ under assumptions that $f$ has non-zero Lyapunov exponents for Lebesgue almost every point and that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-09-20T21:27:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37C40",
      "28D05",
      "37D30"
    ],
    "keywords": [
      "partially hyperbolic systems",
      "positive lyapunov exponents",
      "ergodic lyapunov maximizing observable measure",
      "non-uniformly expanding",
      "dominated splitting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}