{
  "id": "1910.12095",
  "version": "v1",
  "published": "2019-10-26T16:10:16.000Z",
  "updated": "2019-10-26T16:10:16.000Z",
  "title": "On robust expansiveness for sectional hyperbolic attracting sets",
  "authors": [
    "Vitor Araujo",
    "Junilson Cerqueira"
  ],
  "comment": "35 pages; 06 figures; keywords: sectional-hyperbolicity, robust expansiveness, strong dissipativity, star flow, robust transitivity, robust chaotic, attracting sets",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly expansive non-singular vector field is uniformly hyperbolic; and a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-26T16:10:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37D30",
      "37D50"
    ],
    "keywords": [
      "sectional hyperbolic attracting sets",
      "robust expansiveness",
      "expansive non-singular vector field",
      "low dimensional setting",
      "star vector fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}