The study of purely quantum effects in chaotic systems based on semiclassical theory

我们的研究成果是双重的。首先，通过利用所谓的反可积极限，我们能够调查的拓扑结构的四维Smale马蹄，这是我们的小组提出的第一个通用模型的原始Smale马蹄在二维。我们小组的成员已经证明了二维中的几个重要性质，例如动力学的一致双曲性，目前正在对我们的研究结果进行最后的综合。其次，我最近的发现导致了一个新的计划，使用一组特殊的所谓的同宿轨道作为动力学的骨架，构建混沌轨道的符号描述。期望同宿轨道为我们提供关键信息的建设全球马尔可夫分区的整个集合的混沌轨道。沿着这条研究路线的调查沿着有望为我们提供新的工具，使我们能够探讨通用和复杂的系统，否则无法使用原来的Smale马蹄铁方法。研究结果将被纳入目前正在编写的两篇论文：“四维Henon映射的符号动力学”和“同宿轨道马尔可夫划分的整体构造”。

Our research achievements are twofold. First, by making use of the so called anti-integrable limit, we were able to investigate the topological structure of a four-dimensional Smale horseshoe, which is proposed by our group to be the first generic model of the original Smale horseshoe in two dimensions. Several important properties in two dimensions, such as uniformly hyperbolicity of the dynamics, has already been proven by the members of our group, and a final synthesis of our findings is currently undertaken. Second, my recent discovery has led to a new scheme of constructing the symbolic descriptions of chaotic orbits using a special set of so-called homoclinic orbits as the skeletons of the dynamics. The homoclinic orbits are expected to provide us with critical information on the construction of global Markov partitions for the entire set of the chaotic orbits. Investigations along this line of research is promising to provide us with novel tools that allow us to probe into generic and complicated systems that were otherwise inaccessible using the original Smale horseshoe method. The results will be put into two paper that are currently under preparation: “Symbolic Dynamics of a Four-dimensional Henon Map” and "Global Construction of Markov Partitions with Homoclinic Orbits”.