Conformal Dynamics

DMS-0072312术语共形动力学最近获得了一些通用性，并旨在涵盖真实的和复杂的动力学。这种子区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。 该项目提出了各种目标。第一部分是多项式族连通性轨迹的边界研究。这推广了对Mandelbrot集边界的研究。问题包括边界的度量结构，包括补的Riemann映射的边界行为和调和测度的分布.这部分问题延续了J. Graczyk和提议者正在进行的工作。第二个目标是研究Siegel圆盘的边界。这一领域的主要问题是判断这些边界是否是Jordan曲线。所提出的方法是基于获得的Siegel盘的黎曼映射的信息通过一个上同调方程。第三个问题是在realdynamics和关注的存在性野生吸引子additionalproperties。第四个目标是在某些迭代函数系统下估计不变集的测度和Hausdorff维数。要理解这一研究在科学的广阔视野中的意义，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、生态学等自然科学领域中，系统以模型的形式出现，涉及多个参数. 然而，经常出现一个主导参数，它控制系统的长时间行为。在许多复杂系统的研究中，逻辑家族就是这样出现的。逻辑映射是二次多项式。在复平面上研究它们是有用的，因此复二次多项式进入了画面。这是我们研究的第一个对象。在其他情况下，系统开始表现出准旋转或“旋转”行为。这种情况是模拟Siegeldisks这是我们提出的研究的另一个主题。野生拖拉机是我们的第三个主题，它指的是一种非常不寻常的混乱状态，该系统中两种完全不同的长期行为限制模式并存。 我们还将研究的迭代函数系统的吸引子也是某种分形集的投影。这类问题出现在应用中，人们往往有一个多维分形集，可以通过投影到低维空间进行研究。

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.