Conformal Dynamics

DMS-0072312保形动力学一词最近流行起来，旨在涵盖真实和复杂的动力学。这种子领域的合并是有意义的，因为他们的方法比以往任何时候都显示出更多的重叠。该项目提出了多种目标。第一个是研究多项式族的连通性轨迹的边界。这推广了对Mandelbrot集的边界的研究。问题包括边界的度量结构，包括补的黎曼映射的边界行为和调和测度的分布。问题的这一部分延续了J.Graczyk和提出者正在进行的工作。第二个目标是研究Siegel圆盘的边界。这一地区的主要问题是确定这样的边界是否为约旦曲线。所提出的方法是基于通过上同调方程获得Siegel盘的Riemann映射上的信息。第三个问题是实动力学中的，涉及具有附加性质的野生吸引子的存在性。第四个目标是估计在某些迭代函数系下不变集的测度和Hausdorff维数。要从广泛的科学角度理解这项研究的意义，首先必须认识到一维或共形动力学所起的作用。在物理学、天文学、气象学、经济学等自然科学中，以模型形式出现的系统涉及多个参数。然而，经常出现一个控制系统长时间行为的主导参数。就这样，在许多复杂系统的研究中出现了逻辑家族。逻辑映射是二次多项式。在复平面上研究它们是有用的，因此复二次多项式就进入了图景。这是我们研究的第一个对象。在其他情况下，系统开始表现出准周期或“旋转”行为。这是由Siegeldisks模拟的情况，Siegeldisks是我们建议研究的另一个主题。我们的第三个主题--野生拖拉子--指的是两种完全不同的长期行为限制模式并存的系统的一种非常不寻常的混乱状态。迭代函数系统的吸引子表现为某一分形集的投影。这类问题出现在通常具有多维分形集的应用中，该集可以通过在低维空间上的投影来研究。

DMS-0072312The term conformal dynamics has recently gained some currency and ismeant to cover real and complex dynamics. This merging of sub-areasmakes sense because their methods are showing more overlap than ever. The project suggests a variety of goals. The first is the study ofboundaries of connectedness loci for families of polynomials. Thisgeneralizes the study of the boundary of the Mandelbrot set. Problemsinclude the metric structure of the boundary, including the boundarybehavior of the Riemann map of the complement and the distribution ofthe harmonic measure. This part of the problem continues the on-goingwork by J. Graczyk and the proposer. The second goal is the study ofboundaries of Siegel disks. The main problem in this area is decidingwhether such boundaries are Jordan curves. The proposed approach isbased on gaining information on the Riemann map of the Siegel disk bymeans of a cohomological equation. The third problem is in realdynamics and concerns the existence of wild attractors with additionalproperties. The forth goal is to estimate the measure and Hausdorffdimension of sets invariant under certain iterated function systems. To understand the meaning of this research in the broad perspective ofscience, one has first to realize the role played by one-dimensional,or conformal, dynamics. Systems which appear as models in naturalsciences, such as physics, astronomy, meteorology, economics orecology, involve multiple parameters. However, frequently one dominant parameter emergeswhich controls long time-behavior of the system. In such a way thelogistic family appears in the study of many complicated systems. Thelogistic maps are quadratic polynomials. It is useful to study them onthe complex plane, and thus complex quadratic polynomials enter thepicture. This is the first object of our study. In other situations the system begins to show quasi-periodicor ``rotational'' behavior. This is situation is modeled by Siegeldisks which are another subject of our proposed study. Wildattractors, which are our third subject, refer to a very unusualchaotic regime for a system in which two completely different limitingmodes of long-term behavior co-exist side by side. The attractorfor an iterated function system which we will also investigate appearsas a projection of a certain fractal set. This kind of problem appearsin applications where one often has a multi-dimensional fractal setwhich can be studied by projections onto lower-dimensional spaces.