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 Hölder continuous maps.In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space.The space of -fractional currents is in duality with a class of cochains, -fractional charges, that extend both Whitney's flat cochains and -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 -fractional -currents as metric currents of the snowflaked metric space (R , d ( + ) /( +1) Eucl ).