论文标题
在周期性托里的脆弱性上
On the fragility of periodic tori for families of symplectic twist maps
论文作者
论文摘要
在本文中，我们研究了拉格朗日周期性托里（Tori）的脆弱性，用于$ 2D $维环的符号扭曲图，并证明了完全可以整合的扭曲图。 More specifically, we consider $1$-parameter families of symplectic twist maps $(f_\varepsilon)_{\varepsilon\in \mathbb{R}}$, obtained by perturbing the generating function of an analytic map $f$ by a family of potentials $\{\varepsilon G\}_{\varepsilon\in \ mathbb {r}} $。首先，对于分析性$ g $，对于$（m，n）\ in \ mathbb {z} \ times \ mathbb {n}^*$带有$ m $和$ n $ coprime，我们研究了$ \ varepsilon \ in \ varepsilon \ in \ nathbb {r l l l l l l l l l l v _ v _ vareps的拓扑结构向量$（m，n）$。特别是我们证明，在$ f $的合适的非分类条件下，该集合最多由有限的积分组成。然后，在$ c^2 $电位产生的变形的情况下，我们利用这一点来推断可集成的符号扭转图的刚性结果。我们的分析属于任何维度，是基于对拉格朗日定期托里的几何和动态特性的彻底研究，我们认为这具有其自身利益。
In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the $2d$-dimensional annulus and prove a rigidity result for completely integrable ones. More specifically, we consider $1$-parameter families of symplectic twist maps $(f_\varepsilon)_{\varepsilon\in \mathbb{R}}$, obtained by perturbing the generating function of an analytic map $f$ by a family of potentials $\{\varepsilon G\}_{\varepsilon\in \mathbb{R}}$. Firstly, for an analytic $G$ and for $(m,n)\in \mathbb{Z}\times \mathbb{N}^*$ with $m$ and $n$ coprime, we investigate the topological structure of the set of $\varepsilon\in \mathbb{R}$ for which $f_\varepsilon$ admits a Lagrangian periodic torus of rotation vector $(m,n)$. In particular we prove that, under a suitable non-degeneracy condition on $f$, this set consists of at most finitely many points. Then, we exploit this to deduce a rigidity result for integrable symplectic twist maps, in the case of deformations produced by a $C^2$ potential. Our analysis, which holds in any dimension, is based on a thorough investigation of the geometric and dynamical properties of Lagrangian periodic tori, which we believe is of its own interest.
