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

这个项目位于三个不同的数学领域之间的交界处，即遍历理论、分数几何和公制数论。遍历理论是研究随时间演化的系统的统计性质的理论。这门学科的历史可以追溯到19世纪末，当时亨利·庞加莱开始将混沌的概念正规化。自诞生以来，遍历理论就确立了自己作为数学最重要领域之一的地位。遍历理论的一些突出应用包括Einsiedler，Katok和Lindenstrauss关于Littlewood猜想的结果，Furstenberg对Szmeredi定理的证明，以及关于素数算术级数的Green-Tao定理。分形几何学是研究在任意小尺度上表现出复杂性的形状的学科。尽管它起源于20世纪初，但随着现代计算的到来和伯努瓦·B·曼德尔布罗特创造的分形图像，它本身就建立了一个数学领域。在过去的30年里，分形几何蓬勃发展。它现在是一个久负盛名的领域，在现代数学中扮演着重要的角色。度量数论是致力于研究满足某些算术性质的集合的大小的数学领域。人们可以将这个问题追溯到古埃及人，他们对实现圆周率(3.141)的有理近似很感兴趣。作为一门学科，它在20世纪初因埃米尔·波雷尔关于正态数的开创性工作而声名鹊起。这个项目涉及重叠迭代函数系统及其自相似集和度量。近年来，我们对重叠迭代函数系的理解有了很大的进步。特别是，Mike Hochman、Pablo Shmerkin和Peter Varju的结果极大地提高了我们对自相似集合和度量的行为的理解。这些结果也展示了遍历理论、分形几何和度量数理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的基础上。它们的目的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，并提供迭代函数系统的一种新的有意义的分类。这些目标很重要，因为它们直接攻击了分形几何学中最著名的猜想之一，也因为它们有可能改变我们思考迭代函数系统的方式。除了数学之外，计划中的研究成果有可能直接影响从模拟到数字转换、图像压缩和机器人领域的工业问题。

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.