Fractional currents and Young geometric integration
Philippe Bouafia
TL;DR
This work generalizes geometric measure theory to fractal settings by introducing $\alpha$-fractional currents, a class of flat currents admitting $\alpha$-fractional decompositions into normal pieces. It builds a robust duality with $\alpha$-fractional charges, including Hölder forms, and develops a rough calculus featuring a generalized Stokes theorem, pushforwards by Hölder maps, and a higher-dimensional Young integration via a wedge product. The framework yields a generalized Gauss-Green formula for fractal boundaries and provides a compactness theorem, a deformation theorem, and a metric-currents interpretation on snowflaked spaces, thereby unifying fractal geometry with geometric measure theory and extending classical results (e.g., Brouwer degree regularity) to Hölder and fractional settings.
Abstract
We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by Hölder continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $α$-fractional currents is in duality with a class of cochains, $α$-fractional charges, that extend both Whitney's flat cochains and $α$-Hölder continuous forms. We construct a partially defined wedge product between fractional charges, enabling a generalization of the Young integral to arbitrary dimensions and codimensions. This helps us identify $α$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+α)/(m+1)})$.