A counterexample to the singular Weinstein conjecture

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-11-05 DOI:10.1016/j.aim.2024.109998

Josep Fontana-McNally , Eva Miranda , Cédric Oms , Daniel Peralta-Salas

引用次数: 0

Abstract

In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [20]. Our construction shows that there are b-contact manifolds with no singular periodic orbits and no regular periodic orbits away from Z. We do not know whether there are constructions with no generalized escape orbits whose α and ω-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.

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奇异温斯坦猜想的反例

本文研究了 b-contact 流形上 Reeb 向量场的动力学性质。我们证明，在维度 3 中，所谓奇异周期轨道的数量是可以规定的。这些构造阐明了逸出轨道和奇异周期轨道的一些关键性质，它们在提出韦恩斯坦猜想和汉密尔顿塞弗猜想的奇异对应猜想中起着核心作用。事实上，我们证明了上述构造导致了 [20] 中所述这些猜想的反例。我们的构造表明，存在没有奇异周期轨道和远离 Z 的规则周期轨道的 b-contact 流形。我们不知道是否存在没有广义逸出轨道的构造，其 α 和 ω 极限都位于 Z 上（广义奇异周期轨道）。这就是广义韦恩斯坦猜想的内容。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.