Ergodic Theory, Dynamics and Fractals

遍历理论、动力学和分形

• 批准号: 0968879
• 负责人: Boris Solomyak
• 金额: $ 16.7万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968879&HistoricalAwards=false
• 关键词: Ergodic; Theory; Dynamics; Fractals

In this project, the Principal Investigator will build on his earlier work on substitutions, tiling dynamical systems, Bernoulli convolutions, and related topics, to further advance basic mathematical knowledge. Specific goals include extending the orbit equivalence theory and spectral theory to nonminimal Bratteli-Vershik and substitution systems, characterizing invariant measures for nonprimitive substitution tilings, and developing multifractal analysis of Bernoulli convolutions and related self-affine measures. It is expected that new links and connections between diverse areas will emerge in the course of this investigation. In particular, flows along stable and unstable foliations of pseudo-Anosov maps are linked to substitution systems, nonprimitive tilings give rise to fractals, such as the Sierpinski gasket self-similar tiling and the familiar fractal Sierpinski gasket set, complex rational (e.g., quadratic) and piecewise linear dynamics are sometimes related by a conjugacy. The methods and techniques used will not be limited to those from ergodic theory and dynamics but will also come from other fields, including operator algebras, combinatorics, number theory, probability, and complex analysis.The fields of ergodic theory, dynamical systems, and fractal geometry have their origins, motivation, and applications in many fields outside of mathematics, including statistical and celestial mechanics, population biology, aero- and fluid-dynamics, and materials science. A dynamical system is given by a so-called phase space and some transformation rules that prescribe how states change in time. Here "time" may be discrete or continuous, multidimensional, or more generally, given by a group. Dynamics may be regular or chaotic, or it could exhibit a "mixed" behavior. Many dynamically defined objects turn out to be "fractals." Mathematicians often study simplified models that capture the important features of the physical system in the simplest possible form. This project aims to analyze in depth several existing models, as well as develop new ones. Although it is primarily theoretical in its focus, it will advance basic knowledge in areas with diverse connections and applications. Specifically, Bernoulli convolutions and related fractal constructions have been used in signal and image processing, control theory, and econometrics; substitutions and tilings have many links to computer science and solid-state physics. The PI interacts with scientists and participates in professional meetings of an interdisciplinary nature. In addition, this project will contribute to the development of human resources through training undergraduate and graduate students.

在这个项目中，首席研究员将建立在他以前的工作替代，平铺动力系统，伯努利卷积和相关主题，以进一步推进基本的数学知识。具体目标包括扩展轨道等价理论和谱理论的非最小Bratteli-Vershik和替代系统，表征非原始替代tilings不变的措施，并发展多重分形分析伯努利卷积和相关的自仿射措施。预计在调查过程中，不同领域之间将出现新的联系和联系。特别地，沿着伪Anosov映射的稳定和不稳定叶理的沿着流动与替换系统相关联，非原始平铺产生分形，例如Sierpinski垫片自相似平铺和熟悉的分形Sierpinski垫片集，复杂的理性（例如，二次）和分段线性动力学有时通过共轭性相关。所使用的方法和技术将不仅限于遍历理论和动力学，还将来自其他领域，包括算子代数，组合学，数论，概率和复分析。遍历理论，动力系统和分形几何领域在数学之外的许多领域都有其起源，动机和应用，包括统计和天体力学，人口生物学，空气动力学、流体动力学和材料科学。一个动力系统由一个所谓的相空间和一些规定状态如何随时间变化的变换规则给出。这里的“时间”可以是离散的或连续的，多维的，或者更一般地，由一个组给出。动力学可以是规则的或混乱的，或者它可以表现出“混合”行为。许多动态定义的对象最终都是“分形”。数学家经常研究简化模型，这些模型以最简单的形式捕捉物理系统的重要特征。该项目旨在深入分析现有的几种模型，并开发新的模型。虽然它的重点主要是理论，但它将在具有不同联系和应用的领域推进基础知识。具体来说，伯努利卷积和相关的分形结构已被用于信号和图像处理，控制理论和计量经济学;替代和平铺与计算机科学和固态物理学有许多联系。PI与科学家互动，并参加跨学科性质的专业会议。此外，该项目将通过培训本科生和研究生，促进人力资源开发。