Research in dynamics

动力学研究

• 批准号: 0701140
• 负责人: Keith Burns
• 金额: $ 13.98万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701140&HistoricalAwards=false
• 关键词: Research; dynamics

Burns will be continuing his studies of partially hyperbolic dynamical systems. This area 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 recent work of Burns and Wilkinson, has established ergodicity for a very general class of volume preserving partially hyperbolic systems. This work appears to have pushed current techniques to their limit, but still falls short of a complete understanding of when volume preserving partially hyperbolic systems are ergodic; new ideas are needed. Burns will continue to study this question as well as various special examples of partially hyperbolic dynamical systems, in particular compact group extensions of hyperbolic basic sets. He will also consider geometrical questions about the geodesic which are closely related to the work on partial hyperbolicity.Partially hyperbolic dynamical systems are important because they model phenomena which occur in nature and because, at least in some examples, it is possible to completely understand the mechanisms which create the observed phenomena. Partial hyperbolicity is to be expected in systems which have a fast-slow dynamics in which one aspect of the systems operates on a short time scale and other aspects operate on a longer time scale. There is now a good understanding of how partial hyperbolicity gives rise to highly chaotic behaviour in many situations.

伯恩斯将继续他对部分双曲动力系统的研究。在Pugh和Shub的突破性工作之后，这一领域在过去十年中取得了巨大的进展，他们建立了恒定负曲率表面测地线流的时间一图的保体积扰动的遍历性。一系列的论文，在伯恩斯和威尔金森最近的工作中达到高潮，已经建立了一类非常一般的保体积部分双曲系统的遍历性。这项工作似乎已经将当前的技术推向了极限，但仍然没有完全理解什么时候保持体积的部分双曲系统是遍历的；我们需要新的想法。Burns将继续研究这个问题以及部分双曲动力系统的各种特殊例子，特别是双曲基本集的紧群扩展。他还将考虑与部分双曲密切相关的测地线几何问题。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现象，因为至少在某些例子中，有可能完全理解产生所观察到的现象的机制。在具有快慢动力学的系统中，系统的一个方面在短时间尺度上运行，而其他方面在较长时间尺度上运行，预计会出现部分双曲性。现在人们对部分双曲性如何在许多情况下引起高度混乱的行为有了很好的理解。