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学系统的研究。在Pugh和Shub的突破性工作之后，在过去的十年中，该地区取得了长足的进步，后者确立了用于保留时间扰动的体积的千古性，一图的一图的地图是恒定负曲率表面的地球流动。一系列的论文最终是Burns和Wilkinson的最新作品，已经建立了一个非常一般的体积类别，以保留部分双曲线系统。这项工作似乎将当前技术推到了极限，但仍未完全了解何时保留部分双曲线系统是奇异的。需要新想法。 Burns将继续研究这个问题，以及部分双曲动力学系统的各种特殊示例，特别是紧凑的基本基本集合的组。他还将考虑与地质的几何问题，这些问题与部分双曲线的工作密切相关。在自然界中发生的双曲线动力学系统非常重要，因为它们模拟了自然界中发生的现象，并且至少在某些示例中，可以完全理解产生观察到的现象的机制。在具有快速动态动力学的系统中，应期望部分双曲线，其中系统的一个方面在短时间内运行，而其他方面则在更长的时间范围内运行。现在，人们可以很好地理解部分双曲线在许多情况下如何产生高度混乱的行为。