{
  "id": "2111.01921",
  "version": "v2",
  "published": "2021-11-02T22:23:48.000Z",
  "updated": "2025-01-03T15:45:10.000Z",
  "title": "Borel complexity of the family of attractors for weak IFSs",
  "authors": [
    "Paweł Klinga",
    "Adam Kwela"
  ],
  "categories": [
    "math.DS",
    "math.GN"
  ],
  "abstract": "This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS$^d$ of attractors for weak iterated function systems acting on $[0,1]^d$ (as a subset of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric). We prove that wIFS$^d$ is $G_{\\delta\\sigma}$-hard in $K([0,1]^d)$, for all $d\\in\\mathbb{N}$. In particular, wIFS$^d$ is not $F_{\\sigma\\delta}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_{\\sigma}$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-01-03T15:45:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel complexity",
      "weak ifss",
      "iterated function systems attractors",
      "analytic subset",
      "weak iterated function systems acting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}