Complex Dynamics in Higher Dimension

我们研究了高维复杂动力学的三个方面。1. 我们考虑2维或更高维的双元映射。我们使用复分析和代数几何的工具来分析这些映射的基本性质。特别地，我们研究了如何确定所有维度上的动力度。2. 研究了复杂曲面和3-折叠中自同构的存在性。3.我们研究的是复杂Henon族，它是展示复杂动力学行为的重要模型族，因为许多“观察到的”现象可以在复杂情况下用数学方法证明，所以它很重要。我们将重点讨论半抛物型内爆，这是一种重要的分岔现象。这项工作的更广泛的影响将来自于它与其他数学和物理领域的相互作用。特别是在物理学方面，该研究将对晶格统计力学领域产生影响。在数学方面，该研究应与代数几何产生积极的相互作用。这项研究的结果也应该为复杂分析的工具提供新的见解。

We investigate three aspects of complex dynamics in higher dimension. 1. We consider birational mappings in dimension 2 and higher. We use the tools of complex analysis and algebraic geometry to analyze fundamental properties of these mappings. In particular, we investigate how to determine the dynamical degrees in all dimensions. 2. We investigate the existence of automorphisms in complex surfaces and 3-folds. 3.We work with the complex Henon family, which has served as an important model family to exhibit complicated dynamical behaviors, and has been important because many "observed" phenomena can be proved mathematically in the complex case. We will focus on semi-parabolic implosion, which is an important bifurcation phenomenon.The broader impact of this work will come from its interaction with other areas of mathematics and physics. In particular, for physics, the research will have an impact on the area of lattice statistical mechanics. In mathematics, the research should result in a positive cross-fertilization with algebraic geometry. Results of this research should give new insights, too, into the tools of complex analysis.