{
  "id": "1407.0693",
  "version": "v2",
  "published": "2014-07-02T19:55:36.000Z",
  "updated": "2015-01-10T21:00:43.000Z",
  "title": "Hausdorff dimension in graph matchbox manifolds",
  "authors": [
    "Olga Lukina"
  ],
  "comment": "The introduction and Section 5 rewritten, a few references added",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we study the Hausdorff dimension of a certain class of laminations, called graph matchbox manifolds. These laminations are obtained as suspensions of pseudogroup actions on the space of pointed trees, and can be regarded as generalizations of suspensions of shift spaces. It is well-known that full shifts have positive Hausdorff dimension. We show that the Hausdorff dimension of the space of pointed trees is infinite. One of the applications of this result is to the problem of embeddings of laminations into differentiable foliations of smooth manifolds. To admit such an embedding, a lamination must satisfy at least the following two conditions: first, it must admit a metric and a foliated atlas, such that the generators of the holonomy pseudogroup, associated to the atlas, are bi-Lipschitz maps relative to the metric. Second, it must admit an embedding into a manifold, which is a bi-Lipschitz map. The space of graph matchbox manifolds is an example of a lamination where the first condition is satisfied, and the second one is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of a bi-Lipschitz embedding.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-02T19:55:36.000Z",
      "abstract": "A lamination is a compact connected metric space, where each point has a neighborhood homeomorphic to the product of a Euclidean disc and a totally disconnected space. Given a lamination, one can ask if this lamination can be realised as a subset of a smooth foliated finite-dimensional manifold, so that the leaves of the lamination are contained in the leaves of the foliation of the manifold. More precisely, one asks if there exists a foliated embedding of a given lamination into a smooth foliated manifold by a bi-Lipschitz homeomorphism. Hausdorff dimension provides an obstruction to the existence of such an embedding. In the paper, we study a specific class of laminations, called graph matchbox manifolds, obtained as suspensions of pseudogroup actions on the space of pointed trees. We give examples of such laminations which have infinite Hausdorff dimension of their transversals, and, therefore, cannot be embedded as a subset of a smooth foliation of a finite-dimensional manifold by a bi-Lipschitz homeomorphism.",
      "comment": "16 pages, 1 figure, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-10T21:00:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R30",
      "37C45",
      "37C85",
      "37B99"
    ],
    "keywords": [
      "graph matchbox manifolds",
      "lamination",
      "bi-lipschitz homeomorphism",
      "infinite hausdorff dimension",
      "smooth foliated finite-dimensional manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0693L"
    }
  }
}