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.

在该项目中，主要研究者将继续他的动力学研究，并特别强调部分双曲动力学系统。在Pugh和Shub的突破性工作之后，在过去的十年中，部分双曲线取得了长足的进步。一系列论文，最终是威尔金森的主要研究人员的工作，已建立了一类非常一般的批量保存部分双曲线系统的千古。这些结果似乎已将当前技术推向了极限，但仍未完全了解何时具有部分夸张的夸张系统是奇异的：需要新的想法。该项目将在这个问题上追求新鲜的攻击线。 它还将考虑具有表现双曲线行为的几何性质的各种动力学系统。在自然界中发生的重要现象，以及理解这种系统的数学工具，现在已经发达了。例如，现在已经有了很好的理解，即部分双曲性如何产生高度混乱的行为。研究人员还拥有丰富的具体例子，在这些例子中，可以完全理解产生部分双曲行为的机制。从长远来看，正如该项目希望这样做的那样，提高这些知识可能对科学和工程具有重要意义。