Mañé精确磁拉格朗日集的一般性质

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI:10.1134/S1560354721030060

Alexandre Rocha

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引用次数: 0

摘要

设$M$为封闭流形$L$为精确磁拉格朗日量。本文证明了$H^{1}\left(M;\mathbb{R}\right)$的残差集$\mathcal{G}$的存在，使得具有${\widetilde{\mathcal{M}}}\left(c\right)$支持唯一遍历测度的性质$${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$在精确磁拉格朗日族中是一般的。我们还证明了，对于一个固定上同调类$c$，存在一个精确磁拉格朗日残集，使得当这个唯一度量被支持在一个周期轨道上时，这个轨道是双曲的，并且它的稳定流形和不稳定流形横向相交。这个结果是对在[6]中证明的一个类似定理的厌恶。

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Generic Properties of Mañé’s Set of Exact Magnetic Lagrangians

Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we prove that there exists a residual set $\mathcal{G}$ of $H^{1}\left(M;\mathbb{R}\right)$ such that the property

$${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$

with ${\widetilde{\mathcal{M}}}\left(c\right)$ supporting a uniquely ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove that, for a fixed cohomology class $c$, there exists a residual set of exact magnetic Lagrangians such that, when this unique measure is supported on a periodic orbit, this orbit is hyperbolic and its stable and unstable manifolds intersect transversally. This result is a version of an analogous theorem, for Tonelli Lagrangians, proven in [6].

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来源期刊

Regular and Chaotic Dynamics 物理-力学

CiteScore

2.50

自引率

7.10%

发文量

审稿时长

>12 weeks

期刊介绍： Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.