Complex Dynamics in Higher Dimension

这个高维复杂动力学领域的数学研究项目有三个组成部分。第一个组件研究2维及更高维度上的双向映射的属性。贝德福德和他的合作者将使用复杂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，他们将研究如何确定所有维度的动力度。这个项目的第二个组成部分研究了正熵自同构在复杂曲面和3折叠中的存在性。第三部分着重于动力系统的复杂Henon族，它是一个重要的模型族。这些动力系统用非常简单的多项式表示，但它们表现出复杂的动力行为。它们很重要，因为许多观察到的现象实际上可以用数学来证明。这个数学研究项目与混沌理论的研究领域有关，它在物理学、工程学、经济学和生物学等多个学科中都有应用。混沌理论中的一个重要现象是所谓的蝴蝶效应，它说明了给定动力系统对初始条件的不稳定依赖。在这个项目中研究的Henon函数已经成为具有简单公式但表现出蝴蝶效应的转换的肥沃试验场。双空间图的研究在晶格统计力学（LSM）中有着广泛的应用，它研究的是像晶体这样具有晶格结构的材料。LSM提供了一个公式，根据晶格中原子之间的各种相互作用能量来确定这种材料的能量和温度。这个公式很难计算，但在许多情况下，基本的对称性导致了大量的简化。这些对称性是由本项目中所研究的两种地图给出的，而简化的程度是由复杂程度给出的。

This mathematics research project in the area of complex dynamics in higher dimension has three components. The first component investigates the properties of birational mappings in dimension 2 and higher. Bedford and his collaborators will use tools from complex analysis and algebraic geometry to analyze fundamental properties of these mappings. In particular, they will investigate how to determine the dynamical degrees in all dimensions. The second component of this project investigates the existence of automorphisms of positive entropy in complex surfaces and 3-folds. The third component is focused on the complex Henon family of dynamical systems, which has served as an important model family. These dynamical systems are represented by quite simple polynomials, but they exhibit complicated dynamical behaviors. They have been important because many observed phenomena can actually be proved mathematically.This mathematics research project is related to the field of study of chaos theory, which has applications to several disciplines, including physics, engineering, economics and biology. An important phenomenon in chaos theory is the so-called butterfly effect, which illustrates the unstable dependence on initial conditions for a given dynamical system. The Henon functions studied in this project have been a fertile testing ground for transformations that have simple formulas but which exhibit the butterfly effect. The study of birational maps has applications to the field of Lattice Statistical Mechanics (LSM) which considers materials that have a lattice structure like crystals. LSM provides a formula for determining the energy and temperature of such a material in terms of the various interaction energies between the atoms in the lattice. This formula is very difficult to compute, but in many cases there are fundamental symmetries which lead to substantial simplifications. These symmetries are given by birational maps such as those studied in this project, and the amount of simplification is given by the degree complexity.