{
  "id": "1909.02881",
  "version": "v1",
  "published": "2019-09-06T13:03:58.000Z",
  "updated": "2019-09-06T13:03:58.000Z",
  "title": "Shadowing, Internal Chain Transitivity and $α$-limit sets",
  "authors": [
    "Chris Good",
    "Jonathan Meddaugh",
    "Joel Mitchell"
  ],
  "comment": "19 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f \\colon X \\to X$ be a continuous map on a compact metric space $X$ and let $\\alpha_f$, $\\omega_f$ and $ICT_f$ denote the set of $\\alpha$-limit sets, $\\omega$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescribed accuracy) by both the $\\alpha$-limit set and the $\\omega$-limit set of a full-trajectory. Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$ is equal to both the $\\alpha$-limit set and the $\\omega$-limit set of a full-trajectory. In particular this means that shadowing guarantees that $\\overline{\\alpha_f}=\\overline{\\omega_f}=ICT(f)$ (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of c-expansivity entails $\\alpha_f=\\omega_f=ICT(f)$. We progress by introducing novel variants of shadowing which we use to characterise both maps for which $\\overline{\\alpha_f}=ICT(f)$ and maps for which $\\alpha_f=ICT(f)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-06T13:03:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B99",
      "54H20"
    ],
    "keywords": [
      "limit set",
      "internal chain transitivity",
      "compact metric space",
      "closed internally chain transitive sets",
      "nonempty closed internally chain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}