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.打算继续她之前的工作，为塞弗特猜想提供一个真正的分析反例。 P.I. 引入的结构的灵活性。并通过与 Greg Kuperberg 的联合工作进一步开发，允许在 3 维球体上扩展已经很大的平滑非周期流列表。 P.I.计划研究没有 3 维及更高维紧凑叶子以及更高维叶状结构的动力系统的各种特性，以及此类叶状结构的最小集合。尽管有部分结果，但仍不知道每个轨道密集的 3 维球面上是否存在流动。 不存在这种流动的断言被称为戈特沙尔克猜想。 P.I.期望回答一些有关 3 球体上存在最小流量的问题。 “根据毛球定理，不可能将毛球上的所有毛发弄平......这个定理解释了为什么在地球上的任何时刻，水平风速为零。尽管毛球定理很久以前就被证明了，它的高维表亲已经抵制住了攻击，其中最臭名昭著的是海德堡大学的赫伯特·塞弗特 (Herbert Seifert) 在 1950 年提出的问题。奥本大学的克里斯蒂娜·库珀伯格刚刚宣布了令人惊讶的答案……注定会改变高维动力学的面貌。” 高维中的毛球，作者：Ian Stewart，《新科学家》，1993 年 11 月 23 日，第 18 页。Seifert 证明，在某些条件下，3 维球体上的非奇异矢量场具有周期性轨道。 3 球面上不存在非周期矢量场的说法（塞弗特猜想）一直没有得到解决，直到 1974 年 P.A.Schweitzer 的反例（C-1 级，由 J.Harrison 改进为 C-2）为止。 由 P.I. 构建的矢量场回答Seifert的问题比前面的例子要顺利得多——C-无穷大，甚至是实解析的。 与往常一样，新方法为理论研究提出了更多问题和更多可能性，在这种情况下还提出了计算机模拟。