Research in dynamics

动力学研究

基本信息

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

项目摘要

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，这个领域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时间图的体积保持扰动的遍历性。伯恩斯和威尔金森最近发表的一系列论文已经为一类非常一般的体积保持部分双曲系统建立了遍历性。这项工作似乎将当前技术推向了极限，但仍然未能完全理解体积保持部分双曲线系统何时是遍历的；需要新的想法。伯恩斯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。他还将考虑有关测地线的几何问题，这些问题与部分双曲性的工作密切相关。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，并且因为至少在某些例子中，可以完全理解产生观察到的现象的机制。在具有快-慢动态的系统中，预计会出现部分双曲性，其中系统的一方面在短时间尺度上运行，而其他方面在较长的时间尺度上运行。现在对于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。