论文标题
一对分布的动态：双格拉吉亚结构及其在（CO）切线束上的延长和樱桃流动
Dynamic of Pair of some Distributions: Bi-lagrangian structure and its prolongations on the (co)tangent bundles, and Cherry flow
论文作者
论文摘要
我们考虑一个Bi-lagrangian歧管$（M，ω，\ Mathcal {f} _ {1}，\ Mathcal {f} _ {2}）$。也就是说，$ω$是$ m $上的2型，封闭式和非分级（称为symbleticen形式），$（\ Mathcal {f} _ {1}，\ Mathcal {f} _ {2}）$是一对横向lagrangian for the Symplectic carpecorold $（m）$（m）。在这种情况下，$（ω，\ Mathcal {f} _ {1}，\ Mathcal {f} _ {2}）$是$ m $的Bi-lagrangian结构。在本文中，我们以不同的方式延长了其切线捆绑包$ tm $的$ m $和cotangent bundle $ tm $ $ m $的双重结构。结果，在$ m $的双元结构上的一些动态可以延长为$ tm $和$ tm $和$ t^{**} m $的动力学。观察到，一对横向向量场在2-torus $ \ mathbb {t}^2 = \ mathbb {s}^1 \ times \ times \ mathbb {s}^1 $ endowed symplectic形式定义了$ \ mathbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb {t}^2 $的结构上。这激发了我们的好奇心。通过研究$ \ mathbb {t}^2 $上的矢量字段的动态，我们发现可以通过一对矢量字段生成一些带有平坦件（称为樱桃地图）的圆形图（称为樱桃地图）。此外，其矢量字段集合上的一组差异性$ \ mathbb {t}^2 $的推向前进动作在生成的樱桃地图上引起了共轭动作。
We consider a bi-Lagrangian manifold $(M,ω,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $ω$ 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 $(M,ω)$. In this case, $(ω, \mathcal{F}_{1},\mathcal{F}_{2})$ 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.
