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

该项目位于三个不同的数学领域之间的接口，即遍历理论，分形几何和度量数论。遍历理论研究的是随时间演化的系统的统计特性。这门学科的历史可以追溯到19世纪末，当时亨利·庞加莱（Henri Poincare）开始正式提出混沌的概念。自创立以来，遍历理论已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einsedler， Katok和Lindenstrauss关于Littlewood猜想的结果，Furstenberg对Szemeredi定理的证明，以及Green-Tao定理关于素数的等差数列。分形几何是研究在任意小尺度下表现出复杂性的形状。尽管它起源于20世纪初，但直到现代计算的出现和Benoit B. Mandelbrot产生的分形图像才使它成为一个独立的数学领域。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，在现代数学中起着重要的作用。度量数论是研究满足一定算术性质的集合的大小的数学领域。这个主题可以追溯到古埃及人，他们对圆周率（3.141…）的有理近似值很感兴趣。20世纪初，随着埃米尔·博雷尔（Emile Borel）关于正数的开创性工作，它作为一门学科开始崭露头角。本课题研究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认识取得了巨大的进展。特别是，Mike Hochman、Pablo Shmerkin和Peter Varju的研究结果极大地提高了我们对自相似集合和测度行为的理解。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果之上。他们的目标是描述在最近发现的极端行为发生的环境中自相似集和度量的行为，并提供一种新的有意义的迭代函数系统分类。这些目标很重要，因为它们直接攻击了分形几何中最著名的猜想之一，因为它们有可能改变我们对迭代函数系统的思考方式。展望数学之外，计划的研究成果有可能直接影响模拟到数字转换、图像压缩和机器人技术领域的工业问题。

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.