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，利用所谓的反积分极限，我们能够研究四维小马蹄铁的拓扑结构，这是我们小组提出的第一个二维原始小马蹄铁的一般模型。我们小组的成员已经证明了二维中的几个重要性质，如动力学的均匀双曲性，目前正在对我们的发现进行最后的综合。其次，我最近的发现导致了一个新的方案，构建混沌轨道的符号描述，使用一组特殊的所谓的同斜轨道作为动力学的骨架。期望同斜轨道为我们提供构造整个混沌轨道集的全局马尔可夫划分的关键信息。沿着这条研究路线进行的调查有望为我们提供新颖的工具，使我们能够探索使用原始的小马蹄铁方法无法进入的通用和复杂系统。研究结果将发表在目前正在准备的两篇论文中：“四维Henon Map的符号动力学”和“具有同斜轨道的Markov分区的全局构造”。

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”.