{
  "id": "1708.04464",
  "version": "v1",
  "published": "2017-08-15T11:50:43.000Z",
  "updated": "2017-08-15T11:50:43.000Z",
  "title": "Dynamics on the space of 2-lattices in 3-space",
  "authors": [
    "Oliver Sargent",
    "Uri Shapira"
  ],
  "comment": "Submitted. 47 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the dynamics of $SL_3(\\mathbb{R})$ and its subgroups on the homogeneous space $X$ consisting of homothety classes of rank-2 discrete subgroups of $\\mathbb{R}^3$. We focus on the case where the acting group is Zariski dense in either $SL_3(\\mathbb{R})$ or $SO(2,1)(\\mathbb{R})$. Using techniques of Benoist and Quint we prove that for a compactly supported probability measure $\\mu$ on $SL_3(\\mathbb{R})$ whose support generates a group which is Zariski dense in $SL_3(\\mathbb{R})$, there exists a unique $\\mu$-stationary probability measure on $X$. When the Zariski closure is $SO(2,1)(\\mathbb{R})$ we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in $X$. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-15T11:50:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zariski dense",
      "natural open problems",
      "dichotomy regarding stationary measures",
      "stationary probability measure",
      "discrete subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}