Complex Dynamics in Higher Dimension

本项目将研究高维复杂动力学的三个方面：(1)我们考虑平面上真实的、双向的映射，我们将展示如何将复杂解析和代数几何思想应用于这些映射的动力学。(2)研究了矩阵空间上的“逆”映射。我们将确定这个映射的复杂程度，因为它作用于矩阵的各个子空间。(3)我们将研究复杂Henon族，它已经成为展示复杂动力学行为的重要模型族，并且因为许多观察到的现象实际上可以在复杂情况下得到证明而变得重要。我们将研究这个科中的某些抛物线分岔现象。这项工作涉及动力系统和复杂分析领域。更具体地说，动态集合的几何形状与发生在集合上的动态行为有关，这是复分析中一个有趣的研究对象。(2)中所研究的映射是由晶格统计力学中的某些基本对称性引起的。(3)中复杂Henon家族的部分研究应该会影响量子力学隧道的“复杂”方法。因此，(2)和(3)中的工作将对数学物理产生影响。

0601965 BedfordAbstractThis project will investigate three aspects of complex dynamics in higher dimension: (1) We consider real, birational mappings of the plane, and we will show how complex-analytic and algebraic-geometric ideas can be brought to bear on the dynamics of these maps. (2) We study an "inversion" mapping on the space of matrices. We will determine the degree complexities of this map as it acts on various subspaces of matrices. (3) We will 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 actually be proved in the complex case. We will investigate certain parabolic bifurcation phenomena within this family.This work involves the areas of Dynamical Systems and Complex Analysis. More specifically, the geometric shape of the dynamical set is connected with the dynamical behavior that takes place on the set, which is an interesting object of study within Complex Analysis. The mapping studied in (2) arises from certain fundamental symmetries in Lattice Statistical Mechanics. And part of the study of the complex Henon family in (3) should impact a "complex" approach to Quantum Mechanical Tunneling. Thus the work in (2) and (3) will have impact on Mathematical Physics.