{
  "id": "2204.09219",
  "version": "v1",
  "published": "2022-04-20T04:37:43.000Z",
  "updated": "2022-04-20T04:37:43.000Z",
  "title": "An intersection problem of graph-directed attractors and an application",
  "authors": [
    "Huo-Jun Ruan",
    "Jian-Ci Xiao"
  ],
  "comment": "24 pages, 4 figures",
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $(K_1,\\ldots,K_n)$ be a Cantor type graph-directed attractors in $\\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion of whether $K_i\\cap K_j$ is empty for every pair of $1\\leq i,j\\leq n$. Moreover, the emptiness can be checked by examining only a finite number of the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all $d$-dimensional generalized Sierpi\\'nski sponges of which the corresponding IFSs are allowed to have rotational and reflectional components.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-20T04:37:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "54A05"
    ],
    "keywords": [
      "intersection problem",
      "application",
      "cantor type graph-directed attractors",
      "attractors geometric approximations",
      "dimensional generalized sierpinski sponges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}