Dynamics of partially hyperbolic systems and geodesic flows

在这个项目中，首席研究员将继续他在动力学方面的研究，特别是对部分双曲动力系统的研究。在Pugh和Shub的突破性工作之后的过去十年里，部分双曲性有了很大的进展，他们建立了常负曲率曲面测地线流的时间一映射的体积守恒扰动的遍历性。在Wilkinson的主要研究人员的工作中，一系列论文建立了一类非常一般的部分双曲型保容系统的遍历性。这些结果似乎已经将当前的技术推向了极限，但仍然没有提供对体积保持部分双曲系统何时是遍历的完整理解：需要新的想法。这个项目将在这个问题上采取这样的新路线。它还将考虑呈现双曲行为的各种几何性质的动力系统。部分双曲动力系统之所以重要，是因为它们模拟了自然界中发生的重大现象，而且理解这类系统的数学工具到目前为止已经很发达了。例如，现在已经很好地理解了局部双曲性是如何导致高度混乱的行为的。研究人员还拥有一组丰富的具体例子，在这些例子中，有可能完全理解造成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，提高这种知识可能会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.