Research in dynamics

伯恩斯将继续他的研究部分双曲动力系统。这一领域已经看到了巨大的进展，在过去的十年后，突破性的工作Pugh和Shub，谁建立遍历体积保持扰动的时间一地图的测地线流的表面的恒定负曲率。一系列的文件，最终在最近的工作伯恩斯和威尔金森，已经建立了遍历一类非常普遍的体积保持部分双曲系统。这项工作似乎已经推动了目前的技术，以他们的极限，但仍然福尔斯短的体积保持部分双曲系统是遍历的一个完整的理解，需要新的想法。伯恩斯将继续研究这个问题，以及各种特殊的例子，部分双曲动力系统，特别是紧凑组扩展的双曲基本集。他还将考虑几何问题的测地线是密切相关的工作部分hyperbolicity.Partially双曲动力系统是重要的，因为他们的模型现象发生在自然界中，因为，至少在某些例子中，它是可能的，完全理解的机制，创造了观察到的现象。部分双曲性是预期在系统中，具有一个快-慢的动态，其中一个方面的系统操作在一个短的时间尺度和其他方面操作在一个较长的时间尺度。现在有一个很好的理解如何部分双曲引起高度混乱的行为在许多情况下。

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.