{
  "id": "1403.7338",
  "version": "v2",
  "published": "2014-03-28T10:51:38.000Z",
  "updated": "2015-04-28T21:40:12.000Z",
  "title": "Oseledec multiplicative ergodic theorem for laminations",
  "authors": [
    "Viet-Anh Nguyen"
  ],
  "comment": "ix+172 pages. Accepted for publication in Memoirs of the American Mathematical Society. In this second version, several arguments are rewritten with more explanations, and the whole article is thoroughly revised. Moreover, assertion (iii) in the Second Main Theorem has been improved",
  "categories": [
    "math.DS",
    "math.CV",
    "math.DG",
    "math.GT"
  ],
  "abstract": "Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles, we establish effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained. The main ingredients of our method are the theory of Brownian motion, the analysis of the heat diffusions on Riemannian manifolds, the ergodic theory in discrete dynamics and a geometric study of laminations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-28T10:51:38.000Z",
      "abstract": "Given a lamination endowed with a Riemannian metric we introduce the notion of a multiplicative cocycle. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles we establish effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained. The main ingredients of our method are the theory of Brownian motion, the analysis of the heat diffusions on Riemannian manifolds, the ergodic theory in discrete dynamics and a geometric study of laminations.",
      "comment": "146 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-28T21:40:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "57R30",
      "58J35",
      "58J65",
      "60J65"
    ],
    "keywords": [
      "oseledec multiplicative ergodic theorem",
      "lamination",
      "lyapunov exponents",
      "riemannian metric",
      "harmonic probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 172,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7338N"
    }
  }
}