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

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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”.