Dynamics on Bi-Lagrangian Structures and Cherry maps

TL;DR

This paper studies the dynamics of bi-Lagrangian structures on manifolds, especially on the 2-dimensional torus, revealing how lifted structures relate to Cherry maps and their conjugation actions.

Contribution

It introduces methods to prolong bi-Lagrangian structures and connects Cherry vector fields with Cherry maps, including defining linear connections for these maps.

Findings

Lifted bi-Lagrangian structures can be affine in some cases.

Cherry vector fields induce conjugation actions on Cherry maps.

Linear connections for Cherry maps are characterized.

Abstract

We consider a bi-Lagrangian structure $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ on a manifold $M$, that is, $(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-Lagrangian manifold. We prolong bi-Lagrangian structures on $M$, and lift a dynamic on its tangent and cotangent bundles in different ways. In some cases, we show that the lifted structures are affine. In the case of the 2-dimensional torus, we find that an extension of the same dynamic on pairs of so-called Cherry vector fields induces a conjugation action on a subset of Cherry maps (circle maps with a flat). Additionally, we define the linear connections for certain Cherry maps.