2022/05/25 by Bertuel Tangue Ndawa, Ndawa, Bertuel Tangue
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2205.12915
openalex publication_date 2022/05/25 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
We consider a bi-Lagrangian manifold (M,ω,F1,F2). That is, ω is a 2-form, closed and non-degenerate (called symplectic form) on M, and (F1,F2) is a pair of transversal Lagrangian foliations on the symplectic manifold (M,ω). In this case, (ω, F1,F2) 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 \mathbbT2=\mathbbS1×\mathbbS1 endowed with a symplectic form defines a bi-Lagrangian structure on \mathbbT2. This sparked our curiosity. By studying the dynamic of pairs of vector fields on \mathbbT2, 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 \mathbbT2 on the set of its vector fields induces a conjugation action on the set of generated Cherry maps.