A dimension drop phenomenon of fractal cubes
General Topology
2020-10-27 v1
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.
Keywords
Cite
@article{arxiv.2010.12890,
title = {A dimension drop phenomenon of fractal cubes},
author = {Liangyi Huang and Hui Rao},
journal= {arXiv preprint arXiv:2010.12890},
year = {2020}
}