Geometry and Measure in Complex Dynamics

DMS-9803541这个项目的重点是几何结构和丰富的某些不变集出现在非双曲动力学。 主要关注的是Fatou组件的规律性，Hausdorff维数的Julia集，全纯可移除的Julia集，遍历性的不变措施，庞加莱级数的收敛性，和丰富的原型系统。 Collet-Eckmann条件给出了有理映射Julia集的Fatou分支和HD 2的保持器正则性。 该项目旨在将Collet-Eckmann映射的结果推广到所谓的“可求和条件”所给出的更灵活的设置，该条件仅需要沿着临界轨道的多项式增长（可能有驯服的振荡）。 Mandelbrot集边界的外部地址给出了复二次多项式混沌动力学轨迹的自然参数化。 本文的主要目的是研究调和测度在Mandelbrot集的边界上的分布，它将决定Mandelbrot集的1外视图。 该项目的最后一部分涉及椭圆动力学。 研究了Siegel圆和赫尔曼环的正则性。 更一般地说，给定平面上的一条非光滑Jordan曲线，人们可以研究这条曲线上的解析动力学所受的约束。在非线性光滑动力系统中，一维真实的复映射理论占有特殊的地位。 计算的可访问性使得一维映射对于建模和严格研究非双曲系统的性质特别有用，这些系统在更高维度中可能难以接近。 计算机模拟有助于预测这种系统的长期行为和稳定性。 具体的例子来自物理学、生物学和经济学。 也许是最著名的一维模型，其中描述了一个孤立的人口与有限的食物资源的演变，在这个项目中进行了研究（复杂化后）。 许多其他重要的现象，如过渡到混沌，吸引子，分叉轨迹可以通过“分形”几何描述。 分形集具有复杂的结构，通常是理解潜在动力学的关键。 该项目的主要目标之一是探索混沌区域并估计其分形参数。

DMS-9803541The focus of this project is the geometric structure and abundance of certain invariant sets which arise in non-hyperbolic dynamics. Of primary concern are the regularity of Fatou components, the Hausdorff dimension of Julia sets, holomorphic removability of Julia sets, ergodic properties of invariant measures, convergence of the Poincare series, and abundance of prototype systems. The Collet-Eckmann condition yields Holder regularity of Fatou components and HD2 for Julia sets of rational maps. This project aims to generalize results about Collet-Eckmann maps to the more flexible setting given by the so called `summability condition,' which requires only a polynomial growth (with possible tame oscillations) along critical orbits. External addresses of the boundary of the Mandelbrot set give a natural parameterization of the locus of chaotic dynamics of complex quadratic polynomials. The main objective here is to study a distribution of harmonic measure on the boundary of the Mandelbrot set, which will determine the 1outside view' of the Mandelbrot set. The last part of the project concerns elliptic dynamics. The regularity of Siegel disks and Herman rings is considered. More generally, given a non-smooth Jordan curve on the plane, one can investigate constraints imposed on the analytic dynamics on this curve.Among non-linear smooth dynamical systems, the theory of one-dimensional real and complex maps occupies a special place. Computational accessibility makes one-dimensional maps especially useful to model and study rigorously properties of non-hyperbolic systems which may be difficult to approach in higher dimensions. Computer simulations help to predict the long time behavior and stability of such systems. Particular examples come from physics, biology, and economics. Perhaps the most famous one-dimensional model, in which the evolution of an isolated population with limited food resources is described, is studied in this project (after complexification). Many other important phenomena like transition to chaos, attractors, and bifurcation loci can be described through `fractal' geometry. Fractal sets with their complicated structure are often crucial in understanding the underlying dynamics. One of the main objectives of this project is to explore chaotic regions and estimate their fractal parameters.