Ergodic solenoids and generalized currents

TL;DR

This paper introduces the concept of solenoids as laminated spaces, explores their ergodic properties, and defines generalized currents extending Ruelle-Sullivan currents for immersions into smooth manifolds.

Contribution

It presents a novel framework for solenoids as laminated spaces and generalizes currents associated with their immersions, expanding the mathematical understanding of these structures.

Findings

Defined ergodic and uniquely ergodic solenoids.

Established a generalized notion of currents for solenoid immersions.

Extended Ruelle-Sullivan currents to a broader class of spaces.

Abstract

We introduce the concept of solenoid as an abstract laminated space. We do a thorough study of solenoids, leading to the notion of ergodic and uniquely ergodic solenoids. We define generalized currents associated with immersions of oriented solenoids with a transversal measure into smooth manifolds, generalizing Ruelle-Sullivan currents.