Problems of Complex Analysis Arising in Complex Dynamics

抽象的。复动力学中的复分析问题本研究将复分析方法应用于复域中动力系统的几个问题。对复平面C的多项式映射的动力学进行了深入的研究，得出了一个对自然界中的任何系统都是一个美丽而引人注目的模型的理论。二维复空间C^2上多项式映射的动力学应该产生一个美观与实用相等的理论。本课题的工作包括三个部分：1.第一个目标是研究拟双曲映象，这是一个比传统的(一致)双曲性条件更一般且更容易验证的条件。同时，拟双曲映象具有伸缩性质，这使得它们以类似于双曲映象的方式被理解。相应的动力学集将被证明具有有界几何。第二个目标是将这些复方法应用于实平面的多项式微分同态的研究。这将提供一种新的方法，它将给出纯真实方法不容易获得的结果。第三个目标是了解参数空间的性质，即所有映射的空间。将特别关注连通轨迹，它是曼德尔布罗特集的高维类似物。在每一种情况下，方法论都涉及到从复杂分析的数学技术的进一步发展：解析函数和变量、调和测量、多势理论和电流的几何理论。

Abstract. Problems of Complex Analysis Arising in Complex DynamicsThe research of this proposal applies the methods of complex analysis toseveral problems of dynamical systems in the complex domain. The dynamicsof a polynomial map of the complex plane C has been studied in depth andhas produced a theory which is both a beautiful and compelling model formany systems in nature. The dynamics of polynomial mappings oftwo-dimensional complex space C^2 should produce a theory with equalbeauty and utility. The work of this project includes three parts:1. The first objective is to study mappings which are quasi-hyperbolic, acondition which is more general and will be more easily verified than themore traditional condition of (uniform) hyperbolicity. At the same time,the quasi-hyperbolic mappings have expansion/contraction properties whichmakes themunderstandable in a way similar to hyperbolic mappings. The correspondingdynamical sets will be shown to have bounded geometry.2. A second objective is to apply these complex methods to the study ofpolynomial difffeomorphisms of the real plane. This will give a newapproach that will give results not easily obtained with purely real methods.3. The third objective is to understand the nature of parameter space, thespace of all mappings. A particular focus will be on the connectednesslocus, which is a higher-dimensional analogue of the Mandelbrot set.In each of these cases, the methodology involves the further development ofmathematical techniques from complex analysis: analytic functions andvarieties, harmonic measure, pluri-potential theory, and the geometrictheory of currents.