Fractional currents and Young geometric integration

We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by H\"older 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-H\"older 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 (Rᵈ, dEucl(m+α)/(m+1)).