Fractals and Ergodic Theory

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，旨在发现新现象和不同领域之间的新联系。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，可以是任何数字，而不仅仅是整数。 它们广泛应用于数学、物理科学和工程学中，用于对复杂性质的随机和确定性对象进行建模。 分形的数学使用几何测度理论，动力系统理论和其他领域，目的是以严格的方式理解它们的精细结构。 遍历理论是动力系统理论的一个分支，主要研究保测度变换。 这种转换可以被可视化，例如，作为揉面团的过程，或混合不可压缩的流体。它与许多其他领域密切相关，其中包括统计物理学，概率论，数论和组合学。在这个项目中，PI将调查，特别是，分形特性的某些类别的动力系统是目前的极大兴趣。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。 这些包括无限伯努利卷积，这已经研究了近80年，随机连分数，和Furstenberg平稳措施。研究者打算在M. Hochman和P. Shmerkin，并应用加法组合学和傅立叶分析的技术和方法，以获得关于维数，绝对连续性和密度性质的尖锐结果。 另一个研究方向是关于替代动力系统的谱性质和悬浮流，重点是连续或混合谱的系统。PI将研究谱测度何时是纯奇异的，并研究它们的定量性质，特别是维数和Hoelder指数。最近与A。Bufetov揭示了意想不到的联系，这些问题与理论的伯努利卷积。 这个项目的一个重要目标是扩展光谱研究的替代目前感兴趣的其他系统，特别是，间隔交换变换和翻译流的表面上的属大于一。

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.