Ergodic Theory, Dynamics and Fractals

在这个项目中，首席研究人员将在他早期关于代换、瓦片动力系统、伯努利卷积和相关主题的工作的基础上，进一步推进基本的数学知识。具体目标包括将轨道等价理论和谱理论推广到非极小Bratteli-Vershik系统和置换系统，刻画非本原置换瓦片的不变测度，发展Bernoulli卷积和相关自仿射测度的多重分形分析。预计在这次调查过程中，不同地区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的叶层的流动被链接到替换系统，非原始的瓦片产生了分形，例如Sierpinski垫片自相似瓦片和熟悉的分形Sierpinski垫片集，复有理(例如，二次)和分段线性动力学有时通过共轭联系在一起。所使用的方法和技术将不局限于遍历理论和动力学的方法和技术，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的领域有其起源、动机和在数学以外的许多领域的应用，包括统计和天体力学、人口生物学、空气和流体动力学以及材料科学。动力学系统是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的。这里的“时间”可以是离散的或连续的、多维的，或者更一般地，由一组人给出。动力学可能是规则的，也可能是混沌的，也可能表现出“混合”的行为。许多动态定义的对象原来都是“分形图”。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个模型，并开发新的模型。虽然它的重点主要是理论，但它将促进具有不同联系和应用的领域的基本知识。具体地说，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；代换和平铺与计算机科学和固态物理有许多联系。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来促进人力资源的开发。

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.