Dynamic of Pair of some Distributions: Bi-lagrangian structure and its prolongations on the (co)tangent bundles, and Cherry flow

We consider a bi-Lagrangian manifold $(M,\omega,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $\omega$ is a 2-form, closed and non-degenerate (called symplectic form) on $M$, and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations on the symplectic manifold $(M,\omega)$. In this case, $(\omega, \mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-Lagrangian structure on $M$. In this paper, we prolong a bi-Lagrangian structure on $M$ on its tangent bundle $TM$ and its cotangent bundle $T^{*}M$ in different ways. As a consequence some dynamics on the bi-Lagrangian structure of $M$ can be prolonged as dynamics on the bi-Lagrangian structure of $TM$ and $T^{*}M$. Observe that a pair of transversal vector fields without singularity on the 2-torus $\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$ endowed with a symplectic form defines a bi-Lagrangian structure on $\mathbb{T}^2$. This sparked our curiosity. By studying the dynamic of pairs of vector fields on $\mathbb{T}^2$, we found that some circle maps with a flat piece (called Cherry maps) can be generated by a pair of vector fields. Moreover, the push forward action of the set of diffeomorphisms $\mathbb{T}^2$ on the set of its vector fields induces a conjugation action on the set of generated Cherry maps.