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.

摘要：P.I。打算继续她以前的工作，这导致了Seifert猜想的真实分析反例。 P.I.引入的结构的灵活性并在与格雷格·库珀伯格（Greg Kuperberg）的联合作品中进一步发展，可以扩展已经在3维球体上的平稳上的高度流动量表。 P.I.计划研究动态系统的各种特性，而没有紧凑的叶子在维度3及更高和更高维的叶子中，以及此类叶子的最低限度集。尽管存在部分结果，但仍不知道3维球体上是否有流量。 没有这种流动的断言被称为Gottschalk猜想。 P.I.期望回答有关三个球体上最小流量存在的一些问题。 “根据多毛球定理，不可能使所有毛球上的所有头发平滑。...此定理解释了为什么在地球上任何某个地方的某个地方，水平风速为零。虽然毛茸茸的球定理是在很久以前证明的，它很久以前就可以抗拒了最不可思议的攻击。在一个问题上，是一个问题。海德伯格（Heidelberg）。 1993年11月23日，新科学家伊恩·斯图尔特（Ian Stewart）在高维度上的毛茸茸的球证明，在某些条件下，在某些条件下，在三维球体上的非敏感矢量场具有周期性的轨道。直到1974年的P.A.Schweitzer的反例（C-1级，J.Harrison改进到C-Harrison的C-Harrison），尚未解决三个球员猜想的陈述，该声明仍未解决。 P.I.构建的矢量场回答Seifert的问题比以前的示例更顺畅：C-Infinity，甚至是真正的分析。 与往常一样，新方法提出了更多的问题和更多的理论研究可能性，在这种情况下，还提出了计算机模拟。