Mathematical Sciences: Problems of Complex Analysis Arising in Complex Dynamics

本文的研究将复分析方法应用于复域动力系统的若干问题。C的多项式映射的动力学已经得到了深入的研究，并产生了一个理论，它既是一个美丽的和引人注目的模型，许多系统的性质。主要重点是对C^2的多项式微分同态的两部分研究。第一部分涉及Julia集合所连接的映射：分析动态集合J的结构和参数空间（一个高维Mandelbrot空间）的结构。第二部分涉及拟双曲C^2的微分同态：这些映射在无穷小水平上不均匀展开，但在固定（有限）水平上具有均匀展开和有界几何。这项工作涉及复杂分析中的几种数学技术：谐波测量、多势理论和电流几何。许多物理过程被建模为动态系统：系统随着时间的推移演变为特定规则的迭代（重复）。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。该模型最有趣的特征之一是系统的行为依赖于参数的方式（周期倍级联到混沌行为）。当我们转到（看起来更复杂的）复杂系统时，这一点就更清楚了。复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。由此产生的参数（分岔）空间是Mandelbrot集合，它已被证明具有“全称”属性。我们考虑了两个复变量的多项式映射的动力学问题，这是在考虑两个变量的类似行为时得到的结果。我们的研究旨在确定在两个复杂的维度曼德布洛特的模拟，并确定其相关的动态特性。

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.