{
  "id": "2007.10883",
  "version": "v1",
  "published": "2020-07-21T15:10:12.000Z",
  "updated": "2020-07-21T15:10:12.000Z",
  "title": "On backward attractors of interval maps",
  "authors": [
    "Jana Hantáková",
    "Samuel Roth"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Special $\\alpha$-limit sets ($s\\alpha$-limit sets) combine together all accumulation points of all backward orbit branches of a point $x$ under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of $s\\alpha$-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting center that completely characterizes which interval maps have all $s\\alpha$-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic $\\omega$-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of $s\\alpha$-limit sets to the dynamics within them. For example, we show that the isolated points in a $s\\alpha$-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the $s\\alpha$-limit set is nowhere dense. Moreover, we show that $s\\alpha$-limit sets in the interval are always both $F_\\sigma$ and $G_\\delta$. Finally, since $s\\alpha$-limit sets need not be closed, we propose a new notion of $\\beta$-limit sets to serve as backward attractors. The $\\beta$-limit set of $x$ is the smallest closed set to which all backward orbit branches of $x$ converge, and it coincides with the closure of the $s\\alpha$-limit set. At the end of the paper we suggest several new problems about backward attractors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-21T15:10:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37B20",
      "26A18"
    ],
    "keywords": [
      "limit set",
      "interval map",
      "backward attractors",
      "backward orbit branches",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}