{
  "id": "1810.08324",
  "version": "v1",
  "published": "2018-10-19T01:00:31.000Z",
  "updated": "2018-10-19T01:00:31.000Z",
  "title": "On the structure of $λ$-Cantor set with overlaps",
  "authors": [
    "Karma Dajani",
    "Derong Kong",
    "Yuanyuan Yao"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Given $\\lambda\\in(0, 1)$, let $E_\\lambda$ be the self-similar set generated by the iterated function system $\\{x/3,(x+\\lambda)/3,(x+2)/3\\}$. Then $E_\\lambda$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_\\lambda$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_\\lambda$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_\\lambda$ and show that the spectrum of $E_\\lambda$ vanishes if and only if $\\lambda$ is irrational.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-19T01:00:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cantor set",
      "self-similar set",
      "finite triadic codings",
      "totally self-similar",
      "concept first"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}