Hausdorff dimension in graph matchbox manifolds

TL;DR

This paper investigates the Hausdorff and box dimensions of invariant subsets in a pseudogroup dynamical system related to pointed trees, revealing infinite Hausdorff dimension and implications for embedding laminations into smooth manifolds.

Contribution

It demonstrates that the Hausdorff dimension of the space of pointed trees is infinite and explores how this affects the embeddability of laminations into differentiable foliations.

Findings

Hausdorff dimension of the space of pointed trees is infinite.

Dense sets of invariant subsets with non-equal Hausdorff and box dimensions.

Hausdorff dimension acts as an obstruction to bi-Lipschitz embedding of laminations.

Abstract

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 this space is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in this space. We apply our results to the problem of embedding laminations into differentiable foliations of smooth manifolds. One of necessary conditions for the existence of such an embedding is that the lamination must admit a bi-Lipschitz embedding into a manifold. A suspension of the pseudogroup action on the space of pointed graphs gives an example where this condition is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of such a bi-Lipschitz embedding.