Dimension and Dynamics

这个项目的目标是研究非线性迭代函数系统的自相似度量和集、自仿射集和吸引子。重点讨论了符号编码非唯一性时的“重叠”情况，以及与之密切相关的高维非共形情况。将考虑双曲型和抛物型系统。主题包括维度计算，吸引子的拓扑性质的研究，以及自然测度在吸引子上的光滑性性质。本课程将运用符号动力学与遍历理论、数论与几何测度论、调和与复变分析等方法。动力系统描述了广泛的现象，如行星和小行星的运动，人口的上升和下降，天气和股票市场。其中许多系统表现出“混沌”行为，并具有复杂形状和结构的“吸引子”。这种结构通常可以用潜在的“自相似”来解释，这意味着物体在所有尺度上看起来都很相似。吸引子的“分维”是区分不同性质系统的基本特征之一，内部的连通性和存在性等拓扑性质也是非常重要的。我们对这些复杂物体的了解还远远不够，这项研究的目的是在一些模型案例中对它们进行调查。此外，为此而发展的理论也适用于其他领域，如计算机图像识别。

ABSTRACTSolomyak The goal of this project is to investigate self-similar measures and sets, self-affine sets, and attractors of non-linear iterated function systems. The main emphasis is on the "overlapping" case when the symbolic coding is non-unique, and the closely related higher-dimensional non-conformal case. Hyperbolic and parabolic systems will be considered. The topics includedimension computation, the study of topological properties of attractors, and smoothness properties of natural measures on them. The methods of symbolic dynamics and ergodic theory, number theory and geometric measure theory, harmonic and complex analysis will be used. Dynamical systems describe a wide range of phenomena,such as the motion of planets and asteroids, the rise and fall of populations, the weather, and the stock market. Many of these systems exhibit "chaotic" behavior and have "attractors" of intricate shapes and structure. This structure can often be explained by an underlying "self-similarity" which means thatthe object looks similar at all scales. The "fractal dimension" of the attractor is one of the basic characteristics which can be used to distinguish systems of different nature.Topological properties, such as connectedness and presence of the interior are also very important. We are still veryfar from complete understanding of these complex objects, and this study aims to investigate them in a number of model cases. Moreover, the theory developed for this is relevant inother areas, such as computer image recognition.