{
  "id": "1303.6023",
  "version": "v2",
  "published": "2013-03-25T03:52:41.000Z",
  "updated": "2015-11-08T13:07:31.000Z",
  "title": "Expanding curves in $\\mathrm{T}^1(\\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces",
  "authors": [
    "Lei Yang"
  ],
  "comment": "15 pages. The paper is rewritten according to referee's suggestions",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $H = \\mathrm{SO}(n,1)$ and $A =\\{a(t) : t \\in \\mathbb{R}\\}$ be a maximal $\\mathbb{R}$-split Cartan subgroup of $H$. Let $G$ be a Lie group containing $H$ and $\\Gamma$ be a lattice of $G$. Let $x = g\\Gamma \\in G/\\Gamma$ be a point of $G/\\Gamma$ such that its $H$-orbit $Hx$ is dense in $G/\\Gamma$. Let $\\phi: I= [a,b] \\rightarrow H$ be an analytic curve, then $\\phi(I)x$ gives an analytic curve in $G/\\Gamma$. In this article, we will prove the following result: if $\\phi(I)$ satisfies some explicit geometric condition, then $a(t)\\phi(I)x$ tends to be equidistributed in $G/\\Gamma$ as $t \\rightarrow \\infty$. It answers the first question asked by Shah in ~\\cite{Shah_1} and generalizes the main result of that paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-25T03:52:41.000Z",
      "title": "Limit distributions of curves in SO(n,1) acting on homogeneous spaces under geodesic flows",
      "abstract": "Let $G$ be a Lie group and $\\Gamma$ be a lattice of $G$, i.e., $G/\\Gamma$ admits a finite $G$-invariant measure. We consider the left actions of SO(n,1) on $G/\\Gamma$. For a compact segment of analytic curve on SO(n,1) $\\phi: I -> SO(n,1)$ and $x = g\\Gamma \\in G/\\Gamma$, $\\phi(I)x$ gives a curve in the space, we assume that the orbit SO(n,1)x is dense in $G/\\Gamma$. In this article, we consider the limit distributions of such curves under geodesic flow. Since SO(n,1)/SO(n-1) can be identified as the unit tangent bundle of the universal hyperbolic $n$-space, and there is a visual map sending every point in the unit bundle to the ideal boundary sphere of hyperbolic space. It is shown that if the visual map doesn't send the curve into a proper subsphere of the ideal boundary, then under geodesic flow the orbit $\\phi(I)x$ gets asymptotically equidistributed on $G/\\Gamma$. This problem was proposed by Nimish Shah in one of his paper and is a generalization of the main result in that paper. The proof borrows the main technique from that paper but needs some new observations on the representations of SL(2,R).",
      "comment": "16 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-08T13:07:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "22E40"
    ],
    "keywords": [
      "geodesic flow",
      "limit distributions",
      "homogeneous spaces",
      "visual map",
      "ideal boundary sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.6023Y"
    }
  }
}