{
  "id": "2501.11476",
  "version": "v1",
  "published": "2025-01-20T13:09:08.000Z",
  "updated": "2025-01-20T13:09:08.000Z",
  "title": "On recurrence sets for toral endomorphisms",
  "authors": [
    "Zhangnan Hu",
    "Tomas Persson"
  ],
  "comment": "25 pages, 2 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $A$ be a $2\\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus $\\mathbb{T}^2=\\mathbb{R}^2/\\mathbb{Z}^2$, induced by $A$. Given $\\tau>0$, let \\[ R_\\tau =\\{\\, x\\in \\mathbb{T}^2 : T^nx\\in B(x,e^{-n\\tau})~\\mathrm{infinitely ~many}~n\\in\\mathbb{N} \\,\\}. \\] We calculated the Hausdorff dimension of $R_\\tau$, and also prove that $R_\\tau$ has a large intersection property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-20T13:09:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C45",
      "37D20",
      "28A80"
    ],
    "keywords": [
      "toral endomorphisms",
      "recurrence sets",
      "large intersection property",
      "natural endomorphism",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}