Ergodic solenoidal homology: density of ergodic solenoids

TL;DR

This paper proves that currents from uniquely ergodic solenoids are dense among all closed currents, demonstrating the widespread presence of these objects in smooth manifolds.

Contribution

It establishes the density of currents from uniquely ergodic solenoids in the space of all closed currents, extending previous results on their representational capacity.

Findings

Currents from uniquely ergodic solenoids are dense in the space of closed currents.

Any real homology class can be represented by a uniquely ergodic solenoid.

The result confirms the abundance of uniquely ergodic solenoids in smooth manifolds.

Abstract

A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (up to scalars) transversal measure. By the results in arXiv:0910.2913, for any smooth manifold, any real homology class is represented by a uniquely ergodic solenoid. In this paper, we prove that the currents associated to uniquely ergodic solenoids are dense in the space of closed currents, therefore proving the abundance of such objects.