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

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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.