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arXiv · 2010.12890

A dimension drop phenomenon of fractal cubes

Liangyi Huang·Hui Rao
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Abstract

Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdor? dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdor? dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdor? dimension.

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Liangyi Huang, Hui Rao. 2020-10-24. A dimension drop phenomenon of fractal cubes. https://arxiv.org/abs/2010.12890

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