Minimal Dynamical Systems

最小动力系统

• 批准号: 9704558
• 负责人: Krystyna Kuperberg
• 金额: $ 7.14万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-15 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704558&HistoricalAwards=false
• 关键词: Minimal; Dynamical; Systems

Abstract: The P.I. intends to continue her previous work that resulted in a real analytic counterexample to the Seifert Conjecture. The flexibility of the constructions introduced by the P.I. and further developed in a joint work with Greg Kuperberg allows the expansion of the already large list of smooth aperiodic flows on the 3-dimensional sphere. The P.I. plans to investigate various properties of dynamical systems without compact leaves in dimension 3 and higher as well as higher-dimensional foliations, and the minimal sets of such foliations. Is is still not known whether or not there is a flow on the 3-dimensional sphere with every orbit dense, although there are partial results. The assertion that there is no such flow is known as the Gottschalk Conjecture. The P.I. expects to answer some of the questions concerning the existence of minimal flows on the 3-sphere. "According to the hairy ball theorem, it is impossible to smooth down all the hairs on a hairy ball. ... This theorem explains why, for example, at any instant somewhere on Earth the horizontal wind speed is zero. Although the hairy ball theorem was proved long ago, its higher dimensional cousins have resisted attack. The most notorious is the Seifert Conjecture, a question asked in 1950 by Herbert Seifert of the University of Heidelberg. ... The surprising answer, just announced by Krystyna Kuperberg of Auburn University ... destined to change the face of higher-dimensional dynamics." Hairy Balls in Higher Dimensions, by Ian Stewart, New Scientist, 23 November 1993, page 18. Seifert proved that under certain conditions a non-singular vector field on the 3-dimensional sphere has a periodic orbit. The statement that there are no aperiodic vector fields on the 3-sphere, the Seifert Conjecture, remained unsolved until a 1974 counterexample of P.A.Schweitzer (class C-1, improved to C-2 by J.Harrison). The vector field constructed by the P.I. to answer Seifert's question is much smoother than the previous examples- C-infinity, and even real analytic. As usual, new methods raise more questions and more possibilities for theoretical investigations, and in this case also computerized simulations.

翻译后摘要：PI。打算继续她以前的工作，导致了一个真实的分析反例塞弗特猜想。由P.I.引入的结构的灵活性。在与Greg Kuperberg的联合工作中得到进一步发展，使得已经很大的光滑非周期流的列表在三维球体上得到扩展。私家侦探计划研究在3维及更高维以及更高维的无紧叶动力系统的各种性质，以及这些叶的最小集合。目前还不知道是否有一个流动的三维球体上的每一个轨道密集，虽然有部分结果。 不存在这种流动的断言被称为戈特沙尔克猜想。私家侦探期望回答一些关于3-球面上最小流存在性的问题。 “根据毛球定理，不可能把毛球上的所有毛发都弄平。 ... 这个定理解释了为什么，例如，在地球上的某个地方的任何时刻，水平风速为零。 虽然毛球定理在很久以前就被证明了，但它的高维表亲却抵制住了攻击。最臭名昭著的是塞弗特猜想，这是海德堡大学的赫伯特塞弗特在1950年提出的一个问题。 ... 令人惊讶的答案，刚刚由奥本大学的Kaduyna Kuperberg宣布。 注定要改变高维动力学的面貌“Hairy Balls in Higher Dimensions，by Ian Stewart，New Scientist，23 November 1993，page 18. 塞弗特证明了在一定条件下，三维球面上的非奇异向量场有周期轨道。关于三维球面上不存在非周期向量场的陈述，即塞弗特猜想，直到1974年P.A.Schweitzer的一个反例（C-1类，由J.Harrison改进为C-2类）才得以解决。 由P.I.回答塞弗特的问题比前面的例子--C-无穷大，甚至是真实的解析的要顺利得多。 像往常一样，新的方法提出了更多的问题和更多的可能性，理论研究，在这种情况下，也计算机模拟。