Fractals and Ergodic Theory

分形和遍历理论

• 批准号: 1361424
• 负责人: Boris Solomyak
• 金额: $ 17.5万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361424&HistoricalAwards=false
• 关键词: Fractals; Ergodic; Theory

This project is motivated by several interconnected areas of mathematics, with a broad range of applications, and aims at discovering new phenomena and new connections between different fields. Mathematical fractals are sets and measures which exhibit intricate and complicated structure at infinitely many scales - often with some form of self-similarity, and whose "dimension," appropriately defined, can be any number, not just an integer. They are widely used in mathematics, physical sciences, and engineering to model random and deterministic objects of complex nature. The mathematics of fractals uses geometric measure theory, dynamical systems theory, and other fields with the goal of understanding their fine structure in a rigorous way. Ergodic theory is a branch of dynamical systems theory which studies measure-preserving transformations. Such transformations can be visualized, for example, as the process of kneading dough, or mixing an incompressible fluid. It is closely related to many other fields, among them statistical physics, probability, number theory, and combinatorics. In this project, the PI will investigate, in particular, the fractal characteristics of certain classes of dynamical systems that are of great current interest. This award supports the PI's research on self-similar sets and measures and their non-linear analogs, especially in cases of strong overlaps. These include infinite Bernoulli convolutions, which have been studied for almost eighty years, random continued fractions, and Furstenberg stationary measures. The investigator intends to build on the recent progress by M. Hochman and P. Shmerkin and apply the techniques and methods of additive combinatorics and Fourier analysis in order to obtain sharp results on dimension, absolute continuity, and properties of the density. Another research direction is concerned with spectral properties of substitution dynamical systems and suspension flows over them, with an emphasis on systems with continuous or mixed spectrum. The PI will study when the spectral measures are purely singular, and investigate their quantitative properties, in particular, dimensions and Hoelder exponents. Recent collaboration with A. Bufetov revealed unexpected connections of these problems with the theory of Bernoulli convolutions. An important goal of this project is to extend the spectral study of substitutions to other systems of current interest, in particular, to interval exchange transformations and translation flows on surfaces of genus greater than one.

该项目的动机是数学的几个相互关联的领域，具有广泛的应用范围，旨在发现不同领域之间的新现象和新联系。数学分形是在无限多个尺度上表现出错综复杂的结构的集合和测度——通常具有某种形式的自相似性，并且其“维度”，适当定义后，可以是任何数字，而不仅仅是整数。 它们广泛应用于数学、物理科学和工程领域，以对复杂性质的随机和确定性对象进行建模。 分形数学利用几何测度理论、动力系统理论和其他领域，旨在以严格的方式理解其精细结构。 遍历理论是动力系统理论的一个分支，研究测度保持变换。 这种转变可以被形象化，例如，揉捏面团或混合不可压缩流体的过程。它与许多其他领域密切相关，其中包括统计物理、概率、数论和组合学。在这个项目中，PI 将特别研究当前人们非常感兴趣的某些类型的动力系统的分形特征。该奖项支持 PI 对自相似集和测度及其非线性类似物的研究，特别是在强重叠的情况下。 其中包括已经研究了近八十年的无限伯努利卷积、随机连分数和弗斯滕伯格平稳测度。研究人员打算以 M. Hochman 和 P. Shmerkin 的最新进展为基础，应用加法组合学和傅立叶分析的技术和方法，以获得关于密度的维数、绝对连续性和属性的清晰结果。 另一个研究方向涉及替代动力系统和其上的悬浮液的光谱特性，重点是具有连续或混合光谱的系统。 PI 将研究谱测量何时是纯奇异的，并研究它们的定量特性，特别是维数和 Hoelder 指数。最近与 A. Bufetov 的合作揭示了这些问题与伯努利卷积理论的意外联系。 该项目的一个重要目标是将替代谱研究扩展到当前感兴趣的其他系统，特别是大于 1 的属表面上的区间交换变换和平移流。