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维及更高维上的双生映射。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。特别是，我们研究了如何确定所有维度的动态度。2.研究了复杂曲面和3-折叠中自同构的存在性。3.我们研究了复杂的Henon族，它是一个重要的模型族，表现出复杂的动力学行为，并且由于许多“观察到的”现象可以在复杂情况下得到数学证明而变得重要。我们将专注于半抛物线内爆，这是一种重要的分叉现象。这项工作的更广泛影响将来自于它与数学和物理的其他领域的相互作用。特别是对物理学来说，这项研究将对晶格统计力学领域产生影响。在数学上，这项研究应该与代数几何产生积极的交叉。这项研究的结果也应该给复杂分析工具带来新的见解。

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.