Mathematical Sciences: Measures, Dimension and Spectrum

数学科学：测度、维数和谱

• 批准号: 9500744
• 负责人: Boris Solomyak
• 金额: $ 7.5万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500744&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Measures; Dimension; Spectrum

9500744 Solomyak This proposal involves the investigation of Bernoulli convolution measures, self-similar fractal sets and tiling dynamical systems. In the last 15 years, infinite Bernoulli convolutions have become important in smooth dynamics and dimension theory. Using recent progress in understanding these measures, the project will initiate new directions of attack on several interrelated problems, among them a problem of Palis and Takens on the morphology of arithmetic differences of Cantor sets. Tilings are used as models of quasicrystals and a knowledge of their spectrum is important for applications. The project involves developing a framework for the study of general hierarchical tilings which would include the notions of a self-similar tiling. An investigation of spectral properties of old examples and the construction of new examples will be made using ergodic theory and harmonic analysis. Finally an investigation will be made of a well-known example of two-dimensional shift of finite type - the "golden mean" shift. The goal is to compute the entropy of this shift. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Tilings and repeated convolutions can be viewed as dynamical systems. In this way, ergodic theory makes contact with geometry and analysis. This project involves ergodic theory problems along the interface with these areas. ***

9500744 Solomyak 该提案涉及伯努利卷积测度、自相似分形集和平铺动力系统的研究。在过去 15 年中，无限伯努利卷积在平滑动力学和维度理论中变得非常重要。利用最近在理解这些措施方面取得的进展，该项目将对几个相互关联的问题发起新的攻击方向，其中包括关于康托集算术差的形态学的 Palis 和 Takens 问题。平铺被用作准晶体的模型，了解其光谱对于应用非常重要。该项目涉及开发一个用于研究一般分层平铺的框架，其中包括自相似平铺的概念。将使用遍历理论和调和分析对旧示例的光谱特性进行研究并构建新示例。最后将对有限类型二维平移的著名例子——“中庸之道”平移进行研究。目标是计算这种转变的熵。 一般来说，遍历理论涉及理解系统的平均行为，这些系统的动力学过于复杂或混乱，无法在微观细节中遵循。平铺和重复卷积可以被视为动态系统。通过这种方式，遍历理论与几何和分析建立了联系。该项目涉及与这些领域接口的遍历理论问题。 ***