Dynamics of Polynomial Diffeomorphisms

多项式微分同胚的动力学

• 批准号: 9704523
• 负责人: John Smillie
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704523&HistoricalAwards=false
• 关键词: Dynamics; Polynomial; Diffeomorphisms

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.

摘要：该项目涉及在两个复杂的维度的同构的背景下，在一维复杂动力学理论中的一些重要概念的发展类似物。动力学中一个强有力的工具是双曲性。 在一个复杂的维度上，这个属性很容易用临界点的行为来表征。 这个项目的第一部分涉及双曲性和相关性质的两个复杂的变量与期望，这项工作将阐明双曲和非双曲的情况下，双曲性的特征。所提出的刻画涉及到不稳定流形的正则性、不稳定流形与有界轨道集的相交几何以及不稳定流形中绿色函数的增长条件。 第二组问题涉及符号动力学。 在某些情况下，外部光线可以用于参数化“Julia集”。 这在许多情况下给出了一种符号化描述Julia集的新方法。我们提出的问题，哪些象征性的描述可以发生。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。动力学系统是纯数学的一个领域，它处理在研究物理系统的数学模型的长期行为中出现的理论问题。 我们将自己限制在模型中，其状态在任何时候都可以由有限数量的变量来描述。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，可以应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。最近新的技术已经开发出来，使用复杂的分析，但这些技术迄今仅限于一个变量，而典型的动态系统涉及许多变量。所提出的研究旨在建立一个一个复杂的变量的动态和更典型的动力系统，涉及到许多变量之间的联系。