图匹配盒流形的豪斯多夫维数
Hausdorff dimension in graph matchbox manifolds
摘要：
研究了具有伪群作用的点树空间的闭不变子集的Hausdorff维数和盒维数。这个伪群动力系统可以看作是移位空间的推广。证明了该空间的豪斯多夫维数是无穷大的，且具有稠密轨道和不等Hausdorff维数与盒维数的闭不变子集的并在该空间中是稠密的.我们应用我们的结果到问题的嵌入层到可微叶状的光滑流形。这种嵌入存在的必要条件之一是层必须允许双Lipschitz嵌入到流形中。一个在点图空间上的伪群作用的悬置给出了一个不满足这个条件的例子，点树空间的豪斯多夫维数是存在这样一个双Lipschitz嵌入的障碍。
英文摘要：
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.
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