Complex Dynamics and Classification of Algebraic Surfaces

本项目由三个部分组成。第一部分是研究两个变量的多项式映射的动力学。这个部分的主要目的是证明稳定性的拓扑概念等价于双曲性的解析概念。在复杂变动力学和实动力学领域，类似的程序已经导致了大量的活动和许多新的结果。第二部分是利用多能理论的技术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。该组件的主要目标是确定该工具可以应用于高维多项式微分同态的程度，其动力学研究很少。第三部分是根据双曲性的弱概念对代数曲面进行分类。本部分的主要目的是证明从复2空间到代数曲面的秩2全纯映射的存在性等价于复平面的Zariski密像的存在性。这个项目的主要焦点是研究迭代图，这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，它试图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。该过程的交叉部分可以通过与当前项目中研究的非常相似的迭代图来建模。重复过程的一个重要特征是，多次重复过程的参数的微小变化可能导致最终结果的剧烈变化。虽然在某些情况下这可能是理想的行为，但更常见的是更喜欢可靠且相对不受小错误或条件变化影响的系统。因此，很自然地要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.