Ergodic solenoidal homology: Realization theorem

TL;DR

This paper introduces a geometric realization of real homology using generalized currents derived from immersions of ergodic solenoids, providing a new perspective on De Rham's theorem.

Contribution

It establishes a realization theorem for real homology via immersions of minimal uniquely ergodic solenoids, linking geometric currents to algebraic homology.

Findings

Realizes full real homology with generalized currents

Uses minimal uniquely ergodic solenoids for geometric realization

Provides a geometric interpretation of De Rham's theorem

Abstract

We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham's realization of the real homology by only using a restricted geometric subclass of currents.