{
  "id": "0912.5419",
  "version": "v1",
  "published": "2009-12-30T06:13:27.000Z",
  "updated": "2009-12-30T06:13:27.000Z",
  "title": "Breakdown of Normal Hyperbolicity for a Family of Invariant Manifolds with Generalized Lyapunov-Type Numbers Uniformly Bounded below Their Critical Values",
  "authors": [
    "Dennis Guang Yang"
  ],
  "comment": "13 pages, 6 figures, pdfLaTeX",
  "categories": [
    "math.DS"
  ],
  "abstract": "We present three examples to illustrate that in the continuation of a family of normally hyperbolic $C^1$ manifolds, the normal hyperbolicity may break down as the continuation parameter approaches a critical value even though the corresponding generalized Lyapunov-type numbers remain uniformly bounded below their critical values throughout the process. In the first example, a $C^1$ manifold still exists at the critical parameter value, but it is no longer normally hyperbolic. In the other two examples, at the critical parameter value the family of $C^1$ manifolds converges to a nonsmooth invariant set, for which generalized Lyapunov-type numbers are undefined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-30T06:13:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D10"
    ],
    "keywords": [
      "critical value",
      "normal hyperbolicity",
      "invariant manifolds",
      "generalized lyapunov-type numbers remain"
    ],
    "note": {
      "typesetting": "PDFLaTeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.5419Y"
    }
  }
}