Dimension and Dynamics

维度与动力学

• 批准号: 9800786
• 负责人: Boris Solomyak
• 金额: $ 7.69万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800786&HistoricalAwards=false
• 关键词: Dimension; Dynamics

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.

Solomyak 本计画的目标是研究非线性迭代函数系统的自相似测度与集、自仿射集与吸引子。主要强调的是“重叠”的情况下，符号编码是非唯一的，以及密切相关的高维非共形的情况下。将考虑双曲和抛物系统。主要内容包括维数计算、吸引子的拓扑性质和吸引子上的自然测度的光滑性。将使用符号动力学和遍历理论，数论和几何测度理论，调和分析和复分析的方法。 动力系统描述了一系列广泛的现象，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。这些系统中的许多表现出“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“自相似性”来解释，这意味着物体在所有尺度下看起来都是相似的。吸引子的“分形维数”是区分不同性质系统的基本特征之一，其拓扑性质如连通性、内部存在性等也很重要。我们离完全理解这些复杂的物体还很远，本研究的目的是在一些模型案例中研究它们。此外，为此开发的理论与其他领域相关，例如计算机图像识别。