{
  "id": "1705.09593",
  "version": "v1",
  "published": "2017-05-26T14:25:13.000Z",
  "updated": "2017-05-26T14:25:13.000Z",
  "title": "Random matrix products when the top Lyapunov exponent is simple",
  "authors": [
    "Richard Aoun",
    "Yves Guivarc'h"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.GR",
    "math.PR"
  ],
  "abstract": "In the present paper, we treat random matrix products on the general linear group $GL(V)$, where $V$ is a vector space defined on any local field, when the top Lyapunov exponent is simple, without irreducibility assumption. In particular, we show the existence and uniqueness of the stationary measure $\\nu$ on $P(V)$ that is relative to the top Lyapunov exponent and we describe the projective subspace generated by its support. Then, we relate this support to the limit set of the semi-group $T_{\\mu}$ of $GL(V)$ generated by the random walk. Moreover, we show that $\\nu$ has H\\\"older regularity and give some limit theorems concerning the behavior of the random walks: exponential convergence in direction, large deviation estimates of the probability of hitting an hyperplane. These results generalize known ones when $T_{\\mu}$ acts strongly irreducibly and proximally (i-p to abbreviate) on $V$. In particular, when applied to the affine group in the so-called contracting case, the H\\\"older regularity of the stationary measure together with the description of the limit set are new. We mention that we don't use results from the i-p setting; rather we see it as a particular case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-26T14:25:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lyapunov exponent",
      "limit set",
      "random walk",
      "stationary measure",
      "treat random matrix products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}