Mathematical Sciences: Problems of Complex Analysis Arising in Complex Dynamics

数学科学：复动力学中出现的复分析问题

• 批准号: 9706818
• 负责人: Eric Bedford
• 金额: $ 7.49万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-05-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706818&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Complex; Analysis

Abstract Bedford 9706818 The research of this proposal applies to the methods of complex analysis to several problems of dynamical systems in the complex domain. The dynamics of polynomial maps of C has been studied in depth and has produced a theory which is both a beautiful and compelling model for many systems in nature. The principal emphasis is a two-part investigation of polynomial diffeomorphisms of C^2. The first part involves mappings for which the Julia set is connected: to analyze the structure of the dynamical set J and the structure of parameter space (a higher-dimensional Mandelbrot space). The second part involves diffeomorphisms of C^2 which are quasi-hyperbolic: these mappings are not uniformly expanding on the infinitesimal level but have uniform expansion and bounded geometry at a fixed (finite) level . This work involves several mathematical techniques from complex analysis: harmonic measure, pluri-potential theory, and the geometry of currents. Many physical processes are modeled as dynamical systems: systems which evolve over time as the iteration (repetition) of a specific rule. An interesting discovery has been that even if the defining rule is rather simple, the resultant behavior can show surprising complex behavior over time. One of the most intriguing features of this model has been the way the behavior of the system depends on parameters (the period-doubling cascade to chaotic behavior). This is more clearly seen when we pass to the (seemingly more complicated) complex system. The behavior in the complex leads to an interplay between dynamic space and parameter space. The resulting parameter (bifurcation) space is the Mandelbrot set, which has been shown to have "universal" properties. We consider the dynamics of polynomial mappings of two complex variables, which are what is obtained when the analogous behavior is considered in two variables. Our research seeks to identify the analogue of the Mandelbrot in two complex dimensions and to ident ify its associated dynamical properties.

摘要贝德福德9706818这一建议的研究适用于复数域中动力系统的几个问题的复分析方法。C的多项式映射的动力学已被深入研究，并产生了一种理论，对于自然界中的许多系统来说，它既是一个美丽的模型，又是一个引人注目的模型。主要重点是C^2的多项式微分同胚的两部分研究。第一部分涉及与Julia集相连的映射：分析动力集J的结构和参数空间(高维Mandelbrot空间)的结构。第二部分涉及C^2的拟双曲微分同胚：这些映射不是在无穷小水平上一致展开的，而是在固定(有限)水平上具有一致展开和有界几何的。这项工作涉及复数分析中的几种数学技术：谐波测量、多电势理论和电流几何。许多物理过程被建模为动态系统：随着时间的推移作为特定规则的迭代(重复)而演变的系统。一个有趣的发现是，即使定义规则相当简单，随着时间的推移，结果行为也会显示出令人惊讶的复杂行为。该模型最耐人寻味的特征之一是系统的行为依赖于参数的方式(倍周期级联到混沌行为)。当我们谈到(看起来更复杂的)系统时，这一点就更清楚了。综合体中的行为导致了动态空间和参数空间之间的相互作用。由此得到的参数(分支)空间是Mandelbrot集，它已被证明具有“泛”性质。我们考虑两个复变量的多项式映射的动力学，这是当两个变量的相似行为被考虑时得到的。我们的研究试图在两个复数维中识别Mandelbrot的类似物，并识别其相关的动力学性质。