关于高维Lyapunov–Poincaré的一个定理

Q3 Mathematics
V. León, B. Scárdua
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引用次数: 1

摘要

经典的Lyapunov–Poincaré中心定理保证了在二维中心奇异点附近的解析1-形式的第一积分的存在,前提是1-形式的首次喷流是非退化的。基本点是对于给定的1-形式存在解析第一积分。在本文中,我们考虑了两个主要框架的推广:(1)高维余维一的实解析叶理和(2)维二的奇异全纯叶理。所有这一切都与寻找一个准则的问题有关,该准则确保了一个给定余维的具有合适的第一射流的一个芽的解析第一积分的存在。我们的方法包括用全纯叶理来解释中心定理,并遵循Moussu的思想,应用全纯叶理库来获得所需的第一积分。因此,我们能够重新审视Reeb关于精确齐次1-形式的可积扰动的一些经典结果,并在非孤立(横向Morse型扰动)奇点的框架下证明这些结果的版本。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On a Theorem of Lyapunov–Poincaré in Higher Dimensions

The classical Lyapunov–Poincaré center theorem assures the existence of a first integral for an analytic 1-form near a center singularity in dimension two, provided that the first jet of the 1-form is nondegenerate. The basic point is the existence of an analytic first integral for the given 1-form. In this paper, we consider generalizations for two main frameworks: (1) real analytic foliations of codimension one in higher dimension and (2) singular holomorphic foliations in dimension two. All this is related to the problem of finding criteria assuring the existence of analytic first integrals for a given codimension one germ with a suitable first jet. Our approach consists in giving an interpretation of the center theorem in terms of holomorphic foliations and, following an idea of Moussu, apply the holomorphic foliations arsenal to obtain the required first integral. As a consequence we are able to revisit some of Reeb’s classical results on integrable perturbations of exact homogeneous 1-forms, and prove versions of these in the framework of non-isolated (perturbations of transversely Morse type) singularities.

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来源期刊
Arnold Mathematical Journal
Arnold Mathematical Journal Mathematics-Mathematics (all)
CiteScore
1.50
自引率
0.00%
发文量
28
期刊介绍: The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis.  Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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