188 research outputs found
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
Shape of matchbox manifolds
In this work, we develop shape expansions of minimal matchbox manifolds
without holonomy, in terms of branched manifolds formed from their leaves. Our
approach is based on the method of coding the holonomy groups for the foliated
spaces, to define leafwise regions which are transversely stable and are
adapted to the foliation dynamics. Approximations are obtained by collapsing
appropriately chosen neighborhoods onto these regions along a "transverse
Cantor foliation". The existence of the "transverse Cantor foliation" allows us
to generalize standard techniques known for Euclidean and fibered cases to
arbitrary matchbox manifolds with Riemannian leaf geometry and without
holonomy. The transverse Cantor foliations used here are constructed by purely
intrinsic and topological means, as we do not assume that our matchbox
manifolds are embedded into a smooth foliated manifold, or a smooth manifold.Comment: 36 pages. Revision of the earlier version: introduction is rewritten.
Accepted to a special issue of Indagationes Mathematica
- …