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

这个建议是关于耗散系统的长期演化，特别是这些系统最终演化到的最终状态。直觉上，人们会期望一个稳定的平衡状态，或一个周期性振荡的状态。然而，正如现代动力系统理论所揭示的那样，真实的世界要复杂得多。实际上，一个耗散系统可以演化到混沌状态，这种混沌状态就是所谓的“奇异吸引子”，本文对奇异吸引子的一个子类--秩一吸引子进行了数学分析。首先，我们试图了解秩一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，我们试图用数学的严格性来证明这些“奇怪的吸引子”是可观察到的，并且在不同的科学和数学学科中经常遇到。 在现代动力系统理论中，同宿缠结是最重要和最有趣的对象。我们知道马蹄形出现在同宿缠结中，然而，它们只是理论上可测量的相对较小的部分。我们还知道，一般说来，同宿缠结是极其复杂的，也许复杂到无法获得全面的理解。本文主要研究一类特殊的非均匀双曲同宿缠结（秩一吸引子），首先，我们希望在Poincar 'e首次发现这种缠结之后的一百年里，利用许多重要的数学工具（符号动力学、SRB测度和相关的统计性质），对这种同宿缠结的动力学结构有一个全面的了解.其次，我们打算表明，像马蹄铁，这种同宿缠结通常发生在真实的世界，这导致我们的应用。

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.