Some Studies on the Rank One Attractors and the N-Body Problem

这一建议是关于有耗散的系统的长期演化，特别是这些系统最终演化到的最终状态。直觉上，人们会期待一种稳定的平衡状态，或者说周期性振荡的状态。然而，正如现代动力系统理论所揭示的那样，现实世界要复杂得多。实际上，耗散系统可能演化到混沌状态。这些混沌状态就是所谓的“奇异吸引子”。这个建议是基于对一阶吸引子的数学分析，它是奇怪吸引子的一个子类。首先，我们试图了解一阶吸引子的几何和动力学结构，即找到这些混沌行为的基本顺序。其次，我们试图用严谨的数学方法证明，这些“奇怪的吸引子”是可以观察到的，并且经常在不同的科学和数学学科中遇到。在现代动力系统理论中，最重要和最耐人寻味的对象是同向纠缠，这是公认的。我们知道，马蹄铁出现在同宿纠缠中。然而，它们只是理论上可以忽略的相对较小的部分。我们也知道，一般说来，同临床的纠缠是极其复杂的，也许是如此复杂，以至于全面的理解是无望的。这一建议主要是研究一类特殊的非一致双曲同宿纠缠(一阶吸引子)。首先，我们希望利用许多重要的数学工具(符号动力学、SRB度量和一些相关的统计性质)，对这种同宿纠缠的动力学结构有一个全面的理解。这是在Poincar最初发现纠缠的一百年后发展起来的。其次，我们打算证明，就像马蹄铁一样，这种同宿纠缠在现实世界中也普遍存在，这给我们带来了应用。

This proposal is about the long term evolution of systems with dissipation, especially the final states to which these systems evolve at the end. Intuitively,one would expect a state of stable equilibrium, or astate of periodic oscillation. However, as revealed by the modern theory ofdynamical systems, the real world is far morecomplicated. In fact, a dissipative system may evolve to a state of chaos.Thesechaotic statesare the so called ``strange attractors".This proposal is on a mathematical analysis of rank one attractors,a subclass of strange attractors. First we try to understand the geometric anddynamic structure of rank one attractors, i.e., finding the underlining order ofthese chaotic behavior. Second, we try to establish with mathematical rigorthat these ``strange attractors'' are observable and are frequently encounteredin different discipline of science and mathematics. In the modern theory of dynamical systems, it is well acknowledgedthat the most important and intriguing objects are homoclinictangles. We know that horseshoes occur inside a homoclinic tangle.They are, however, only a relatively small part that is measuretheoretically ignorable. We also know that, in general, homoclinictangles are extremely complicated, perhaps so complicated thatcomprehensive understandings are hopeless to acquire. This proposal is mainlyon the study of a specific nonuniformlyhyperbolic homoclinic tangle (rank one attractors).First we hope tooffer a comprehensive understanding of the dynamical structure ofhomoclinic tangles of this kind by using many of the importantmathematical tools (Symbolic dynamics, SRB measures and relatedstatistical properties for a few) developed in the past onehundred years after the initial findings of the tangles byPoincar\'e. Second we intend to show that, like the horseshoes,homoclinic tangles of this kind commonly occur in the real world,which lead us to applications.