{
  "id": "1909.03061",
  "version": "v1",
  "published": "2019-09-06T13:14:50.000Z",
  "updated": "2019-09-06T13:14:50.000Z",
  "title": "Orbital shadowing, $ω$-limit sets and minimality",
  "authors": [
    "Joel Mitchell"
  ],
  "comment": "7 pages. arXiv admin note: text overlap with arXiv:1907.02446",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $X$ be a compact Hausdorff space, with uniformity $\\mathscr{U}$, and let $f \\colon X \\to X$ be a continuous function. For $D \\in \\mathscr{U}$, a $D$-pseudo-orbit is a sequence $(x_i)$ for which $(f(x_i),x_{i+1}) \\in D$ for all indices $i$. In this paper we show that pseudo-orbits trap $\\omega$-limit sets in a neighbourhood of prescribed accuracy after a uniform time period. A consequence of this is a generalisation of a result of Pilyugin et al: every system has the second weak shadowing property. By way of further applications we give a characterisation of minimal systems in terms of pseudo-orbits and show that every minimal system exhibits the strong orbital shadowing property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-06T13:14:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B99",
      "54H20"
    ],
    "keywords": [
      "limit sets",
      "minimality",
      "minimal system",
      "uniform time period",
      "compact hausdorff space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}