{
  "id": "2104.09546",
  "version": "v1",
  "published": "2021-04-19T18:13:05.000Z",
  "updated": "2021-04-19T18:13:05.000Z",
  "title": "Expanding measures: Random walks and rigidity on homogeneous spaces",
  "authors": [
    "Roland Prohaska",
    "Cagri Sert",
    "Ronggang Shi"
  ],
  "comment": "60 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $G$ be a real Lie group, $\\Lambda<G$ a lattice and $H<G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of $H$-expanding measures $\\mu$ on $H$ and, applying recent work of Eskin-Lindenstrauss, prove that $\\mu$-stationary probability measures on $G/\\Lambda$ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that $H$-expanding random walks on $G/\\Lambda$ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\\Lambda$ for $H$-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-19T18:13:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B15",
      "22F30",
      "60G50",
      "37A45",
      "28A80"
    ],
    "keywords": [
      "expanding measures",
      "homogeneous spaces",
      "expanding random walks",
      "construct lyapunov/margulis functions",
      "stationary probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}