Dynamics on Bi-Lagrangian Structures and Cherry maps

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.