Dynamics of Polynomial Diffeomorphisms

摘要：这个项目涉及在二维复数维的微分同胚的背景下，开发一维复动力学理论中的一些重要概念的类比。动力学中的一个强大工具是双曲性。在一个复杂的维度中，这一性质很容易根据临界点的行为来表征。这个项目的第一部分涉及双曲性的刻划和两个复变量中微分同胚的相关性质，期望这项工作将有助于揭示双曲和非双曲的情况。所提出的刻划涉及不稳定流形的正则性、不稳定流形与有界轨道集的交集的几何以及不稳定流形中格林函数的增长条件。第二组问题涉及符号动力学。在某些情况下，外部光线可以用来对‘’Julia Sets‘’进行参数化。这为在许多情况下象征性地描述Julia集提供了一种新的方法。我们提出了哪些符号描述可以出现的问题。第三组问题涉及“Julia集的连通性”的性质和连通性轨迹的性质。动力系统是一个纯数学领域，它研究在研究物理系统的数学模型的长期行为时出现的理论问题。我们将自己限制在模型中，这些模型在任何时候的状态都可以用有限个变量来描述。尽管这些系统可以表现出复杂的混沌行为，但在某些情况下，可以使用一些强大的理论工具来研究它们。另一方面，在许多情况下，这些工具并不适用。最近发展了一些使用复杂分析的新技术，但到目前为止，这些技术仅限于一个变量，而典型的动力系统涉及许多变量。所提出的研究旨在建立一个复变量的动力学与更典型的涉及多个变量的动力系统之间的联系。

Abstract: This project involves developing analogs of some of the important concepts in the theory of one dimensional complex dynamics in the context of diffeomorphisms in two complex dimensions. One powerful tool in dynamics is hyperbolicity. In one complex dimension this property is easily characterized in terms of the behavior of the critical point. The first part of this project involves characterizations of hyperbolicity and related properties for diffeomorphisms in two complex variables with the expectation that this work will shed light both on the hyperbolic and the non-hyperbolic cases. The proposed characterizations involve the regularity of unstable manifolds, the geometry of the intersection of unstable manifolds with the set of bounded orbits and growth conditions on Green functions in unstable manifolds. A second set of questions involve symbolic dynamics. There are situations in which external rays can be used to parametrize ``Julia sets''. This gives a new way of symbolically describing Julia sets in many cases. We raise questions about which symbolic descriptions can occur. A third set of questions involve the property of ``connectivity of the Julia set'' and the properties of the connectivity locus. Dynamical systems is an area of pure mathematics which deals with theoretical questions that arise in studying long term behavior of mathematical models of physical systems. We restrict ourselves to models whose state at any time can be described by a finite number of variables. Even though these systems can display complicated chaotic behavior, there are some powerful theoretical tools that can be applied to study them in some cases. On the other hand there are many cases where these tools do not apply. Recently new techniques have been developed which use complex analysis but these techniques have so far been limited to one variable whereas typical dynamical systems involve many variables. The proposed research aims to establish a link between dynamics of one complex v ariable and the more typical dynamical systems which involve many variables.