{
  "id": "0904.1142",
  "version": "v1",
  "published": "2009-04-07T13:45:22.000Z",
  "updated": "2009-04-07T13:45:22.000Z",
  "title": "On Homoclinic points, Recurrences and Chain recurrences of volume-preserving diffeomorphisms without genericity",
  "authors": [
    "Jaeyoo Choy",
    "Hahng-Yun Chu",
    "Min Kyu Kim"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS",
    "math.SG"
  ],
  "abstract": "Let $M$ be a manifold with a volume form $\\omega$ and $f : M \\to M$ be a diffeomorphism of class $\\mathcal{C}^1$ that preserves $\\omega$. In this paper, we do \\textit{not} assume $f$ is $\\mathcal{C}^1$-generic. We have two main themes in the paper: (1) the chain recurrence; (2) relations among recurrence points, homoclinic points, shadowability and hyperbolicity. For (1) (without assuming $M$ is compact), we have the theorem: if $f$ is Lagrange stable, then $M$ is a chain recurrent set. If $M$ is compact, then the Lagrange-stability is automatic. For (2) (assuming the compactness of $M$), we prove some various implications among notions, such as: (i) the $\\mathcal{C}^1$-stable shadowability equals to the hyperbolicity of $M$; (ii) if a point $p\\in M$ has a recurrence point in the unstable manifold $W^u (p, f)$ and there is no homoclinic point of $p,$ then $f$ is nonshadowable; (iii) if $f$ has the shadowing property and $p$ has a recurrence point in $W^u (p, f),$ then the recurrent point is in the limit set of homoclinic points of $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-07T13:45:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C50",
      "37C29",
      "37D05"
    ],
    "keywords": [
      "homoclinic point",
      "chain recurrence",
      "volume-preserving diffeomorphisms",
      "recurrence point",
      "genericity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.1142C"
    }
  }
}