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维及更高维的双有理映射。 我们使用复变函数分析和代数几何的工具来分析这些映射的基本性质。 特别是，我们研究如何确定在所有维度的动力学程度。2. 研究了复曲面和3-折叠中自同构的存在性。 3.我们研究的是复Henon族，它是一个重要的模型族，表现出复杂的动力学行为，并且很重要，因为许多“观察到的”现象可以在复情况下得到数学证明。 我们将集中讨论半抛物内爆，这是一个重要的分岔现象，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。 特别是对于物理学来说，这项研究将对 格点统计力学领域。 在数学方面，这项研究应该会与代数几何产生积极的交叉影响。 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.