Dynamics of partially hyperbolic systems and geodesic flows

在这个项目中，主要研究者将继续他的研究，特别是在部分双曲动力系统的动力学。部分双曲性在过去的十年里取得了很大的进展，在Pugh和Shub的突破性工作之后，他们建立了常负曲率曲面的测地线流的时间一映射的保体积扰动的遍历性。一系列的文件，最终在工作的主要研究员与威尔金森，已经建立了遍历一类非常普遍的体积保持部分双曲系统。这些结果似乎已经把目前的技术，以他们的极限，但仍然达不到提供一个完整的理解时，体积保持部分双曲系统遍历：新的想法是必要的。这个项目将在这个问题上采取新的攻击路线。 部分双曲动力系统是重要的，因为它们模拟了自然界中发生的重要现象，而且理解这类系统的数学工具到目前为止已经发展得很好。例如，现在对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，提高这方面的知识可能对科学和工程产生重要影响。

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.