Dynamics of partially hyperbolic systems and geodesic flows

部分双曲系统和测地流的动力学

• 批准号: 1001959
• 负责人: Keith Burns
• 金额: $ 16.6万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001959&HistoricalAwards=false
• 关键词: Dynamics; partially; hyperbolic; systems; geodesic

In this project the principal investigator will be continue his studies in dynamics with a particular emphasis on partially hyperbolic dynamical systems. Partial hyperbolicity has seen great progress in the last decade after the breakthrough work of Pugh and Shub, who established ergodicity for volume-preserving perturbations of the time-one map of the geodesic flow of a surface of constant negative curvature. A series of papers, culminating in work of the principal investigator with Wilkinson, has established ergodicity for a very general class of volume-preserving partially hyperbolic systems. These results appear to have pushed current techniques to their limit, but still fall short of providing a complete understanding of when volume-preserving partially hyperbolic systems are ergodic: new ideas are needed. This project will pursue such fresh lines of attack on this question. It will also consider various dynamical systems of a geometrical nature that exhibit hyperbolic behavior.Partially hyperbolic dynamical systems are important because they model significant phenomena that occur in nature and because the mathematical tools for understanding such systems are by now well developed. For example, there is now a good understanding of how partial hyperbolicity gives rise to highly chaotic behavior. Researchers also possess a rich set of concrete examples in which it is possible to understand completely the mechanisms that create the partially hyperbolic behavior. In the long run, improving this knowledge, as this project hopes to do, could have important implications for science and engineering.

在这个项目中，主要研究者将继续他的动力学研究，特别是部分双曲动力系统。在 Pugh 和 Shub 的突破性工作之后，部分双曲性在过去十年中取得了巨大进展，他们建立了恒定负曲率表面的测地流的时间一图的体积保持扰动的遍历性。主要研究者与 Wilkinson 合作的一系列论文已经为一类非常通用的体积保持部分双曲系统建立了遍历性。这些结果似乎已经将当前技术推向了极限，但仍然无法提供对保持体积的部分双曲系统何时遍历的完整理解：需要新的想法。该项目将对这个问题采取新的攻击方式。 它还将考虑表现出双曲行为的各种具有几何性质的动力系统。部分双曲动力系统很重要，因为它们模拟了自然界中发生的重要现象，并且因为用于理解此类系统的数学工具目前已经很发达。例如，现在对于部分双曲性如何引起高度混沌行为有了很好的理解。研究人员还拥有一组丰富的具体例子，可以完全理解产生部分双曲行为的机制。从长远来看，正如该项目所希望的那样，提高这些知识可能会对科学和工程产生重要影响。