{
  "id": "math/0303310",
  "version": "v1",
  "published": "2003-03-25T13:01:53.000Z",
  "updated": "2003-03-25T13:01:53.000Z",
  "title": "Sufficient conditions for robustness of attractors",
  "authors": [
    "C. A. Morales",
    "M. J. Pacifico"
  ],
  "comment": "17 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "A recent problem in dynamics is to determinate whether an attractor $\\Lambda$ of a $C^r$ flow $X$ is $C^r$ robust transitive or not. By {\\em attractor} we mean a transitive set to which all positive orbits close to it converge. An attractor is $C^r$ robust transitive (or {\\em $C^r$ robust} for short) if it exhibits a neighborhood $U$ such that the set $\\cap_{t>0}Y_t(U)$ is transitive for every flow $Y$ $C^r$ close to $X$. We give sufficient conditions for robustness of attractors based on the following definitions. An attractor is {\\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \\cite{MPP}. An attractor is {\\em $C^r$ critically-robust} if it exhibits a neighborhood $U$ such that $\\cap_{t>0}Y_t(U)$ is in the closure of the closed orbits is every flow $Y$ $C^r$ close to $X$. We show that on compact 3-manifolds all $C^r$ critically-robust singular-hyperbolic attractors with only one singularity are $C^r$ robust.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-03-25T13:01:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D30",
      "37D45"
    ],
    "keywords": [
      "sufficient conditions",
      "robustness",
      "critically-robust singular-hyperbolic attractors",
      "volume expanding central direction",
      "robust transitive"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......3310M"
    }
  }
}