An extension of fractal Euler number via persistent homology

TL;DR

This paper extends the fractal Euler number concept using persistent homology and magnitude theory, enabling calculations for complex fractals like Cantor dust and Menger sponge previously not covered.

Contribution

It introduces a modified fractal Euler number leveraging persistent homology, broadening applicability to more complex fractals.

Findings

Calculated the average ph-fractal Euler number for Cantor dust.

Calculated the average ph-fractal Euler number for Menger sponge.

Extended the applicability of fractal Euler numbers to new fractal classes.

Abstract

In the context of geometric measure theory, Llorente-Winter introduced the (average) fractal Euler number as a notion of the Euler characteristic for fractals embedded in Euclidean space. However, the class of fractals to which it is applicable remains very limited. In the present paper, we modify this notion by applying perspectives of persistent homology and partly the theory of magnitude, which have recently come from applied topology and category theory. We then demonstrate concrete calculation of our average ph-fractal Euler number for some classically well-known fractals, especially the Cantor dust and Menger sponge which are excluded from Llorente-Winter's approach.