{
  "id": "1703.07224",
  "version": "v1",
  "published": "2017-03-21T13:57:07.000Z",
  "updated": "2017-03-21T13:57:07.000Z",
  "title": "Pointwise Equidistribution and Translates of Measures on Homogeneous Spaces",
  "authors": [
    "Osama Khalil"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X,\\mathfrak{B},\\mu)$ be a Borel probability space. Let $T_n: X\\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\\nu$ be a probability measure on $X$ such that $\\frac{1}{N}\\sum_{n=1}^N (T_n)_\\ast\\nu \\rightarrow \\mu$ in the weak-$\\ast$ topology. Under general conditions, we show that for $\\nu$ almost every $x\\in X$, the measures $\\frac{1}{N}\\sum_{n=1}^N \\delta_{T_n x}$ get equidistributed towards $\\mu$ if $N$ is restricted to a set of full upper density. We present applications of these results to expanding translates of curves on homogeneous spaces and translates of orbits of symmetric groups. As a corollary, we prove equidistribution of certain sparse orbits of the horocycle flow on quotients of $SL(2,\\mathbb{R})$, starting from every point in almost every direction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-21T13:57:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homogeneous spaces",
      "pointwise equidistribution",
      "translates",
      "borel probability space",
      "full upper density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}