{
  "id": "2310.18971",
  "version": "v1",
  "published": "2023-10-29T10:34:20.000Z",
  "updated": "2023-10-29T10:34:20.000Z",
  "title": "Structures of $R(f)-\\overline{P(f)}$ for graph maps $f$",
  "authors": [
    "Jiehua Mai",
    "Enhui Shi",
    "Kesong Yan",
    "Fanping Zeng"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $G$ be a graph and $f: G\\rightarrow G$ be a continuous map. We establish a structure theorem which describes the structures of the set $R(f)-\\overline{P(f)}$, where $R(f)$ and $P(f)$ are the recurrent point set and the periodic point set of $f$ respectively. Roughly speaking, the set $R(f)-\\overline{P(f)}$ is covered by finitely many pairwise disjoint $f$-invariant open sets $U_{1\\,},\\,\\cdots,\\,U_{n\\,}$; each $U_i$ contains a unique minimal set $W_i$ which absorbs each point of $U_i$; each $W_i$ lies in finitely many pairwise disjoint circles each of which is contained in a connected closed set; all of these connected closed sets are contained in $U_i$ and permutated cyclically by $f$. As applications of the structure theorem, several known results are improved or reproved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-29T10:34:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E25",
      "37B20",
      "37C25",
      "54H20"
    ],
    "keywords": [
      "graph maps",
      "structure theorem",
      "connected closed set",
      "unique minimal set",
      "periodic point set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}