Minimal Dynamical Systems

摘要：P.I.打算继续她之前的工作，这项工作导致了对塞弗特猜想的真正的分析反例。P.I.引入的结构的灵活性，并在与Greg Kuperberg的联合工作中进一步发展，允许在三维球面上扩展已经很大的光滑非周期流动列表。P.I.计划研究没有3维及更高维紧叶的动力系统的各种性质，以及更高维的叶层，以及这些叶层的最小集。尽管已经有了部分结果，但仍然不知道在每个轨道密集的三维球体上是否有流动。不存在这种流动的断言被称为哥沙尔克猜想。P.I.希望回答有关3-球面上存在极小流的一些问题。根据毛球定理，不可能使毛球上的所有毛发变得光滑。例如，这个定理解释了为什么在地球上的任何时刻，毛球的水平风速为零。尽管毛球定理很久以前就被证明了，但它的高维表亲们一直抵制攻击。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……注定要改变高维动力学的面貌。《高维毛球》，作者：Ian Stewart，《新科学家》，1993年11月23日，第18页。Seifert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。关于三维球面上没有非周期矢量场的说法，塞弗特猜想，直到1974年P.A.Schweitzer的反例(C-1类，由J·哈里森改进为C-2类)才被解决。P.I.为回答Seifert的问题而构造的矢量场比前面的例子要平滑得多--C-无穷大，甚至是实解析的。像往常一样，新的方法为理论研究提出了更多的问题和更多的可能性，在这种情况下，也提出了计算机模拟。

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.