{
  "id": "1303.5993",
  "version": "v5",
  "published": "2013-03-24T21:17:44.000Z",
  "updated": "2015-11-08T13:24:24.000Z",
  "title": "Hausdorff dimension of divergent diagonal geodesics on product of finite volume hyperbolic spaces",
  "authors": [
    "Lei Yang"
  ],
  "comment": "29 pages. The paper is rewritten according to referees' suggestions",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this article, we consider the product space of several non-compact finite volume hyperbolic spaces, $V_1, V_2, \\dots , V_k$ of dimension $n$. Let $\\mathrm{T}^1(V_i)$ denote the unit tangent bundle of $V_i$ for each $i=1,\\dots , k$, then for every $(v_1, \\dots , v_k) \\in \\mathrm{T}^1 (V_1) \\times \\cdots \\times \\mathrm{T}^1 (V_k)$, the diagonal geodesic flow $g_t$ is defined by $g_t (v_1, \\dots , v_k) = (g_t v_1, \\dots , g_t v_k)$. And we define $$\\mathfrak{D}_k =\\left\\{ (v_1, \\dots, v_k) \\in \\mathrm{T}^1 (V_1) \\times \\cdots \\times \\mathrm{T}^1 (V_k): g_t(v_1, \\dots, v_k) \\text{ divergent, as } t\\rightarrow \\infty\\right\\}. $$ We will prove that the Hausdorff dimension of $\\mathfrak{D}_k$ is equal to $k(2n-1) - \\frac{n-1}{2}$. This extends the result of Yitwah Cheung ~\\cite{Cheung1}.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-11-18T20:02:12.000Z",
      "title": "Hausdorff dimension of divergent geodesics on product of hyperbolic spaces",
      "abstract": "In this article, we consider the diagonal geodesic flows on product of $k$ copies of non-compact hyperbolic spaces with finite volumes of dimension $n$ and we want to study the points on the product space such that their trajectories under geodesic flows are divergent. We prove that the set of such points has Hausdorff dimension $k(2n-1) -(n-1)/2$. This work is a generalization of a result of Y. Cheung, which deals with the product space of copies of the arithmetic space $SL(2,Z)\\setminus SL(2,R)$. The proof involves some interesting results on the distribution of the orbits of discrete group actions on Furstenberg boundary.",
      "comment": "36 pages, there are some minor changes compared with the previous version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-11-08T13:24:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "37A25",
      "22E40"
    ],
    "keywords": [
      "hausdorff dimension",
      "divergent geodesics",
      "product space",
      "non-compact hyperbolic spaces",
      "discrete group actions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}