Research in dynamics

彭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，这一领域在过去的十年中取得了很大的进展，他们建立了时间的体积保持扰动的遍历性，这是常负曲率曲面的测地线流的一个映射。在Burns和Wilkinson最近的工作中，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体积保持部分双曲系统何时是遍历的；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张。他还将考虑与偏双曲工作密切相关的关于测地线的几何问题。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现象，而且至少在一些例子中，完全可能理解产生所观察到的现象的机制。在具有快-慢动态的系统中，部分双曲性是可以预期的，其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。现在，人们已经很好地理解了局部双曲性如何在许多情况下导致高度混乱的行为。

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.