{
  "id": "2305.02879",
  "version": "v1",
  "published": "2023-05-04T14:44:59.000Z",
  "updated": "2023-05-04T14:44:59.000Z",
  "title": "Stationary probability measures on projective spaces 2: the critical case",
  "authors": [
    "Richard Aoun",
    "Cagri Sert"
  ],
  "comment": "15 pages",
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "In a previous article, given a finite-dimensional real or complex vector space $V$ and a probability measure $\\mu$ on $\\operatorname{PGL}(V)$ with finite first moment, we gave a description of all $\\mu$-stationary probability measures on the projective space $\\operatorname{P}(V)$ in the non-critical (or Lyapunov dominated) case. In the current article, we complete the analysis by providing a full description of the more subtle critical case. Our results demonstrate an algebraic rigidity in this situation. Combining our results with those of Furstenberg--Kifer ('83), Guivarch--Raugi ('07) $\\&$ Benoist--Quint ('14), we deduce a classification of all stationary probability measures on the projective space for i.i.d random matrix products with finite first moment without any algebraic assumption.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-04T14:44:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37H15",
      "60B15",
      "60J05"
    ],
    "keywords": [
      "stationary probability measures",
      "projective space",
      "critical case",
      "finite first moment",
      "complex vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}