{
  "id": "0912.2699",
  "version": "v3",
  "published": "2009-12-14T18:08:32.000Z",
  "updated": "2010-05-04T13:40:47.000Z",
  "title": "Nonuniform Hyperbolicity, Global Dominated Splittings and Generic Properties of Volume-Preserving Diffeomorphisms",
  "authors": [
    "Artur Avila",
    "Jairo Bochi"
  ],
  "comment": "Some corrections were made.",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of points with only non-zero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold $C^r$-generically for any $r \\geq 1$, namely: the continuity of the ergodic decomposition, the persistence of invariant sets, and the $L^1$-continuity of Lyapunov exponents.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-05-04T13:40:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D25",
      "37D30",
      "37C20"
    ],
    "keywords": [
      "global dominated splitting",
      "nonuniform hyperbolicity",
      "generic properties",
      "zero lyapunov exponent",
      "study generic volume-preserving diffeomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2699A"
    }
  }
}