A dimension drop phenomenon of fractal cubes
A dimension drop phenomenon of fractal cubes
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DOI：
10.1016/j.jmaa.2020.124918
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发表时间：
2020-10
影响因子：
1.3
通讯作者：
Liang-yi Huang;H. Rao
中科院分区：
文献类型：
--
作者：
Liang-yi Huang;H. Rao
设一个度量空间。我们引入了e的连通度指标的概念，它是e的非平凡连通分量并集的豪斯多夫维数。我们证明了一个分形立方体的连通指数严格小于它的Hausdorff维数，只要它具有平凡的连通分量。因此连通性指标是一个新的Lipschitz不变量。此外，我们还研究了连通性指数与拓扑Hausdorff维数之间的关系。
LetEbe a metric space. We introduce a notion of connectedness index ofE, which is the Hausdorff dimension of the union of non-trivial connected components ofE. We show that the connectedness index of a fractal cubeEis strictly less than the Hausdorff dimension ofEprovided thatEpossesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdorff dimension.