Persistent Homology and the Upper Box Dimension

TL;DR

This paper introduces a new fractal dimension based on persistent homology, compares it to the upper box dimension, and explores conditions under which they coincide, especially in subsets of Euclidean space.

Contribution

It defines a novel fractal dimension using persistent homology and establishes conditions for its equivalence to the upper box dimension in certain metric spaces.

Findings

The new dimension is comparable to the upper box dimension under specific hypotheses.

For subsets of  with upper box dimension > 1.5, the dimensions coincide.

Results relate to extremal questions about persistent homology intervals in finite point sets.

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.