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

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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.