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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 Lagrangianfoliations on the symplectic manifold $(M,\omega)$. In this case, $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$ is a bi-Lagrangian structure on $M$. In thispaper, 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 somedynamics on the bi-Lagrangian structure of $M$ can be prolonged as dynamics onthe bi-Lagrangian structure of $TM$ and $T^{*}M$. Observe that a pair oftransversal vector fields without singularity on the 2-torus$\mathbb{T}^2=\mathbb{S}^1\times\mathbb{S}^1$ endowed with a symplectic formdefines a bi-Lagrangian structure on $\mathbb{T}^2$. This sparked ourcuriosity. 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 begenerated by a pair of vector fields. Moreover, the push forward action of theset of diffeomorphisms $\mathbb{T}^2$ on the set of its vector fields induces aconjugation action on the set of generated Cherry maps.

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