Fractals and Ergodic Theory

University Of Washington, Seattle WA

Investigators

Abstract

This project is motivated by several interconnected areas of mathematics, with a broad range of applications, and aims at discovering new phenomena and new connections between different fields. Mathematical fractals are sets and measures which exhibit intricate and complicated structure at infinitely many scales - often with some form of self-similarity, and whose "dimension," appropriately defined, can be any number, not just an integer. They are widely used in mathematics, physical sciences, and engineering to model random and deterministic objects of complex nature. The mathematics of fractals uses geometric measure theory, dynamical systems theory, and other fields with the goal of understanding their fine structure in a rigorous way. Ergodic theory is a branch of dynamical systems theory which studies measure-preserving transformations. Such transformations can be visualized, for example, as the process of kneading dough, or mixing an incompressible fluid. It is closely related to many other fields, among them statistical physics, probability, number theory, and combinatorics. In this project, the PI will investigate, in particular, the fractal characteristics of certain classes of dynamical systems that are of great current interest. This award supports the PI's research on self-similar sets and measures and their non-linear analogs, especially in cases of strong overlaps. These include infinite Bernoulli convolutions, which have been studied for almost eighty years, random continued fractions, and Furstenberg stationary measures. The investigator intends to build on the recent progress by M. Hochman and P. Shmerkin and apply the techniques and methods of additive combinatorics and Fourier analysis in order to obtain sharp results on dimension, absolute continuity, and properties of the density. Another research direction is concerned with spectral properties of substitution dynamical systems and suspension flows over them, with an emphasis on systems with continuous or mixed spectrum. The PI will study when the spectral measures are purely singular, and investigate their quantitative properties, in particular, dimensions and Hoelder exponents. Recent collaboration with A. Bufetov revealed unexpected connections of these problems with the theory of Bernoulli convolutions. An important goal of this project is to extend the spectral study of substitutions to other systems of current interest, in particular, to interval exchange transformations and translation flows on surfaces of genus greater than one.