Persistent Homology and the Upper Box Dimension

Benjamin Schweinhart

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arXiv · 1802.00533

Persistent Homology and the Upper Box Dimension

Benjamin Schweinhart

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Abstract

We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of $\mathbb{R}^2$ whose upper box dimension exceeds $1.5.$ These results are related to extremal questions about the number of persistent homology intervals of a set of $n$ points in a metric space.

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Benjamin Schweinhart. 2018-02-02. Persistent Homology and the Upper Box Dimension. https://arxiv.org/abs/1802.00533

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