Complex Dynamics and Classification of Algebraic Surfaces

摘要秃鹰9706018这个项目有三个组成部分。第一个部分是研究两个变量多项式映射的动力学。这一部分的主要目的是证明欧米伽稳定性的拓扑学概念等价于双曲性的解析概念。类似的程序导致了在单复变量动力学和实动力学领域的大量活动和许多新的结果。第二部分是利用多势理论研究三个变量的多项式映射。多势理论是研究一元和二元多项式映射动力学的有效工具。这个部分的主要目的是确定这个工具在多大程度上可以应用于高维多项式微分同胚，其动力学研究很少。第三个组成部分是根据弱双曲性概念对代数曲面的分类。这一部分的主要目的是证明从复2-空间到代数曲面的二阶全纯映射的存在性等价于复平面的Zariski稠密像的存在性。这个项目的主要焦点是研究迭代映射，迭代映射是对重复多次的过程的数学抽象。这种重复的过程在许多不同的领域中都很常见。例如，许多生物过程，如DNA复制，都是重复多次的简单过程。与当前项目更直接相关的是遗传算法的概念，这是一种尝试利用生物繁殖的一些特征，如遗传交叉和竞争，来产生有效的计算机算法。这一过程的交叉部分可以通过迭代地图进行建模，这些地图与当前项目中正在研究的地图非常相似。重复过程的一个重要特征是，重复多次的过程参数的微小变化可能会导致最终结果的剧烈变化。虽然在某些情况下这可能是所需的行为，但更常见的是选择可靠且相对不受小错误或条件变化影响的系统。因此，很自然地会问，哪些过程是稳定的，因为参数的微小变化会导致最终结果的微小变化。解决某些系统的这个问题是当前项目的主要目标。

Abstract Buzzard 9706018 This project has three components. The first component is the study of the dynamics of polynomial maps of two variables. The primary goal of this component is to show that the topological notion of Omega stability is equivalent to the analytic notion of hyperbolicity. Similar programs have led to a great deal of activity and many new results in the fields of one complex variable dynamics and real dynamics. The second component is the study of polynomial mappings of three variables using techniques of pluripotential theory. Pluripotential theory has been an effective tool in the study of the dynamics of polynomial mappings of one and two variables. The main goal of this component is to determine the extent to which this tool can be applied to higher dimensional polynomial diffeomorphisms, whose dynamics have been studied very little. The third component is the classification of algebraic surfaces in terms of a weak notion of hyperbolicity. The main goal of this component is to show that the existence of a rank-two holomorphic map from complex 2-space to an algebraic surface is equivalent to the existence of a Zariski dense image of the complex plane. The main focus of this project is the study of iterated maps, which are mathematical abstractions for processes which are repeated many times. Such repeated processes are commonly found in many diverse fields. For instance, many biological processes, such as DNA replication, are simple procedures which are repeated many times. More directly related to the current project is the notion of a genetic algorithm, which is an attempt to use some of the features of biological reproduction, such as genetic crossover and competition, to produce effective computer algorithms. The crossover part of this procedure can be modeled by iterated maps very similar to those under study in the current project. An important feature of repeated processes is that many times, a slight change in the parameters of a process which is repeated many times can lead to a drastic change in the final outcome. While this may be desirable behavior in some cases, it is more common to prefer systems which are reliable and relatively unaffected by small errors or changes in conditions. Hence it is natural to ask which processes are stable in the sense that a small change in parameters leads to a small change in the final outcome. Resolving this question for some systems is the primary aim of the current project.