Overlapping iterated function systems: New approaches and breaking the super-exponential barrier

重叠迭代函数系统：新方法和打破超指数障碍

• 批准号: EP/W003880/1
• 负责人: Simon Baker
• 金额: $ 34.62万
• 依托单位: Loughborough University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW003880%2F1
• 关键词: Overlapping; iterated; function; systems; New

This project lies at the interface between three distinct areas of mathematics, namely Ergodic Theory, Fractal Geometry, and Metric Number Theory. Ergodic Theory is the study of the statistical properties of systems that evolve with time. The history of this subject dates back to the late 19th century when Henri Poincare began to formalise the notion of chaos. Since its inception Ergodic Theory has established itself as one of the most important fields of Mathematics. Some standout applications of Ergodic Theory include the results of Einsiedler, Katok, and Lindenstrauss on the Littlewood conjecture, Furstenberg's proof of Szemeredi's theorem, and the Green-Tao theorem on arithmetic progressions in the primes. Fractal Geometry is the study of shapes that exhibit complexity at arbitrarily small scales. Despite having its origins in the early 20th century, it took the advent of modern computing and the fractal images produced by Benoit B. Mandelbrot to establish it as a mathematical field in its own right. Over the last 30 years Fractal Geometry has flourished. It is now a well-established field and plays an important role in modern mathematics. Metric Number Theory is the field of mathematics devoted to studying the size of sets satisfying certain arithmetic properties. One can trace this subject back to the ancient Egyptians who were interested in achieving rational approximations to pi (3.141...). As a subject it rose to prominence in the early 20th century with the pioneering work of Emile Borel on normal numbers. This project is concerned with overlapping iterated function systems and their self-similar sets and measures. In recent years tremendous progress has been made in our understanding of overlapping iterated function systems. In particular, the results of Mike Hochman, Pablo Shmerkin, and Peter Varju have significantly improved our understanding of the behaviour of self-similar sets and measures. These results have also exhibited new and deep connections between Ergodic Theory, Fractal Geometry, and Metric Number Theory. The research objectives of this project build upon these achievements. They aim to describe the behaviour of self-similar sets and measures in an environment where a recently discovered extreme behaviour occurs, and to provide a new meaningful classification of iterated function systems. These objectives are important because they directly attack one of the most well-known conjectures in Fractal Geometry, and because they have the potential to transform the way we think about iterated function systems. Looking beyond mathematics, the planned research outputs have the potential to directly impact industrial problems from the fields of analogue to digital conversion, image compression, and robotics.

该项目位于数学的三个不同领域之间的接口，即遍历理论，分形几何和度量数论。遍历理论是研究随时间演化的系统的统计特性的理论。这个主题的历史可以追溯到世纪末，当时亨利·庞加莱开始正式提出混沌的概念。自诞生以来，遍历理论已经成为数学中最重要的领域之一。遍历理论的一些突出应用包括艾因西德勒、卡托克和林登施特劳斯关于利特尔伍德猜想的结果，弗斯滕伯格关于塞梅雷迪定理的证明，以及关于素数算术级数的格林-陶定理。分形几何是研究在任意小尺度上表现出复杂性的形状。尽管它起源于世纪早期，但它是随着现代计算和Benoit B制作的分形图像的出现而出现的。曼德尔布罗特建立它作为一个数学领域在自己的权利。在过去的30年里，分形几何蓬勃发展。它现在是一个成熟的领域，并在现代数学中发挥着重要作用。度量数论（英语：Metric Number Theory）是一个数学领域，致力于研究满足某些算术性质的集合的大小。人们可以将这个问题追溯到古埃及人，他们对实现π（3.141...）的合理近似感兴趣。作为一个主题，它上升到突出在世纪初的开创性工作埃米尔波莱尔正常数。这个项目关注的是重叠迭代函数系统及其自相似集和测度。近年来，我们对重叠迭代函数系统的理解取得了巨大的进展。特别是，Mike Hochman、巴勃罗Shmerkin和Peter Varju的研究结果大大提高了我们对自相似集和测度行为的理解。这些结果也展示了遍历理论，分形几何和度量数论之间新的和深刻的联系。本项目的研究目标建立在这些成就的基础上。他们的目标是描述行为的自相似集和措施的环境中，最近发现的极端行为发生，并提供一个新的有意义的分类迭代函数系统。这些目标很重要，因为它们直接攻击分形几何中最著名的几何之一，并且因为它们有可能改变我们对迭代函数系统的思考方式。在数学之外，计划的研究成果有可能直接影响模数转换、图像压缩和机器人等领域的工业问题。

项目成果

Intrinsic Diophantine approximation for overlapping iterated function systems

重叠迭代函数系统的本征丢番图近似

• DOI: 10.1007/s00208-023-02608-8
• 发表时间: 2023
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Baker S

Spectral gaps and Fourier dimension for self-conformal sets with overlaps

具有重叠的自共形集的谱间隙和傅立叶维数

• DOI: 10.48550/arxiv.2306.01389
• 发表时间: 2023
• 作者: Baker S