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

Abstract

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 $T^{\ast }M$ 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 $T^{\ast }M$. 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.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:2107.05149