Fractals and Ergodic Theory

该项目由几个相互关联的数学领域驱动，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结构的集合和度量，通常具有某种形式的自相似性，其“维度”可以是任何数字，而不仅仅是整数。它们广泛应用于数学、物理科学和工程中，以模拟复杂性质的随机和确定性对象。分形的数学运用了几何测量理论、动力系统理论和其他领域，目的是严谨地理解分形的精细结构。遍历理论是研究保测度变换的动力系统理论的一个分支。这种转换可以可视化，例如，作为揉面团的过程，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、概率论、数论和组合学。在这个项目中，PI将特别研究当前非常感兴趣的某些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80年的无限伯努利卷积、随机连分数和弗斯滕伯格平稳测度。研究者打算在M. Hochman和P. Shmerkin的最新进展的基础上，应用加性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、绝对连续性和性质的清晰结果。另一个研究方向是研究取代动力系统及其上悬浮流的光谱性质，重点研究连续或混合光谱系统。PI将研究谱测度是纯奇异的情况，并研究它们的定量性质，特别是维数和霍尔德指数。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。本项目的一个重要目标是将置换的谱研究扩展到当前感兴趣的其他系统，特别是在大于1的属表面上的区间交换变换和平移流。

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.