Complex Dynamics in Higher Dimension

高维复杂动力学

• 批准号: 0601965
• 负责人: Eric Bedford
• 金额: --
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601965&HistoricalAwards=false
• 关键词: Complex; Dynamics; Higher; Dimension

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.

0601965贝德福德摘要本项目将研究复杂动力学的三个方面， 高维：（1）我们考虑空间的真实的双有理映射， 平面，我们将展示如何复解析和代数几何 想法可以被用来影响这些地图的动态。 (2)我们 研究矩阵空间上的“逆”映射。 我们将 确定该映射的复杂度，因为它作用于各种 矩阵的子空间 (3)我们将与复杂的Henon 家庭，作为一个重要的模范家庭， 复杂的动力学行为，并一直很重要，因为许多 观察到的现象实际上可以在复杂的情况下得到证明。 我们 将研究某些抛物线分叉现象， 家庭。这项工作涉及领域的动力系统和复杂 分析. 更具体地说，动力学的几何形状 集合与发生在 集合，这是复分析中一个有趣的研究对象。 在（2）中研究的映射源于格点统计力学中的某些基本对称性。 在（3）中对复杂Henon家族的部分研究应该会影响量子力学Tunneling的“复杂”方法。 因此，（2）和（3）中的工作将对数学物理产生影响。