{
  "id": "1605.06864",
  "version": "v1",
  "published": "2016-05-23T00:00:20.000Z",
  "updated": "2016-05-23T00:00:20.000Z",
  "title": "On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms",
  "authors": [
    "Jorge Rocha",
    "Paulo Varandas"
  ],
  "comment": "16 pages, 5 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we study $C^1$-structurally stable diffeomorphisms, that is, $C^1$ Axiom A diffeomorphisms with the strong transversality condition. In contrast to the case of dynamics restricted to a hyperbolic basic piece, structurally stable diffeomorphisms are in general not expansive and the conjugacies between $C^1$-close structurally stable diffeomorphisms may be non-unique, even if there are assumed $C^0$-close to the identity. Here we give a necessary and sufficient condition for a structurally stable diffeomorphism to admit a dense subset of points with expansiveness and sensitivity to initial conditions. Morever, we prove that the set of conjugacies between elements in the same conjugacy class is homeomorphic to the $C^0$-centralizer of the dynamics. Finally, we use this fact to deduce that any two $C^1$-close structurally stable diffeomorphismsare conjugated by a unique conjugacy $C^0$-close to the identity if and only if these are Anosov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-23T00:00:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "initial conditions",
      "sensitivity",
      "uniqueness",
      "strong transversality condition",
      "hyperbolic basic piece"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}