Hausdorff dimension in graph matchbox manifolds

In this paper, we study the Hausdorff and the box dimensions of closed invariant subsets of the space of pointed trees, equipped with a pseudogroup action. This pseudogroup dynamical system can be regarded as a generalization of a shift space. We show that the Hausdorff dimension of the space of pointed trees is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in the space of pointed trees. We apply our results to the problem of embedding 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. A suspension of the pseudogroup action on the space of pointed graphs gives 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.Comment: Proof of Theorem 1.1 simplified as compared to the previous version; Sections 5 and 6 contain new result

The topological cyclic Deligne conjecture

Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2-discs operad.Comment: An appendix comparing with the construction by McClure and Smith has been adde