On simultaneous conjugacies of pairs of transverse foliations of the torus

TL;DR

This paper proves that for pairs of transverse minimal topological foliations on the torus, individual conjugacy implies simultaneous conjugacy, establishing a key equivalence in foliation theory.

Contribution

It establishes a necessary and sufficient condition for simultaneous conjugacy of pairs of transverse minimal foliations on the torus.

Findings

Individual conjugacy implies simultaneous conjugacy for the pairs.

The equivalence holds specifically for minimal topological foliations.

Provides a new criterion for foliation conjugacy on the torus.

Abstract

We prove in this note that two pairs of transverse minimal topological foliations of the torus are individually conjugated if, and only if they are simultaneously conjugated.