{
  "id": "2306.10523",
  "version": "v1",
  "published": "2023-06-18T11:01:18.000Z",
  "updated": "2023-06-18T11:01:18.000Z",
  "title": "On Periodic Points in Covering Systems",
  "authors": [
    "Yihan Wang"
  ],
  "comment": "12 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:2205.07225",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "We study a system of intervals $I_1,\\ldots,I_k$ on the real line and a continuous map $f$ with $f(I_1 \\cup I_2 \\cup \\ldots \\cup I_k)\\supseteq I_1 \\cup I_2 \\cup \\ldots \\cup I_k$. It's conjectured that there exists a periodic point of period $\\le k$ in $I_1\\cup \\ldots \\cup I_k$. In this paper, we prove the conjecture by a discretization method and reduce the initial problem to an interesting combinatorial lemma concerning cyclic permutations. We also obtain a non-concentration property of periodic points of small periods in intervals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-18T11:01:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic point",
      "covering systems",
      "combinatorial lemma concerning cyclic permutations",
      "interesting combinatorial lemma concerning cyclic",
      "real line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}