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.