{
  "id": "2402.04810",
  "version": "v1",
  "published": "2024-02-07T12:55:24.000Z",
  "updated": "2024-02-07T12:55:24.000Z",
  "title": "Hausdorff dimension of recurrence sets for matrix transformations of tori",
  "authors": [
    "Zhangnan Hu",
    "Bing Li"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $T\\colon\\mathbb{T}^d\\to \\mathbb{T}^d$, defined by $T x=Ax(\\bmod 1)$, where $A$ is a $d\\times d$ integer matrix with eigenvalues $1<|\\lambda_1|\\le|\\lambda_2|\\le\\dots\\le|\\lambda_d|$. We investigate the Hausdorff dimension of the recurrence set \\[R(\\psi):=\\{x\\in\\mathbb{T}^d\\colon T^nx\\in B(x,\\psi(n)) {\\rm ~for~infinitely~ many~}n\\}\\] for $\\alpha\\ge\\log|\\lambda_d/\\lambda_1|$, where $\\psi$ is a positive decreasing function defined on $\\mathbb{N}$ and its lower order at infinity is $\\alpha=\\liminf\\limits_{n\\to\\infty}\\frac{-\\log \\psi(n)}{n}$. In the case that $A$ is diagonalizable over $\\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-07T12:55:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C45",
      "37B20",
      "28A80"
    ],
    "keywords": [
      "hausdorff dimension",
      "recurrence set",
      "matrix transformations",
      "integer matrix",
      "lower order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}