Hausdorff dimension in graph matchbox manifolds

Olga Lukina

Published 2014-07-02, updated 2015-01-10Version 2

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.