Statistical and Number-Theoretical Properties of Dynamical Systems

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

There are classical results in probability theory that allow one to predict the average long term behavior of random phenomena. The aim of the proposed project is to study to what extent certain classes of deterministic phenomena can be treated as if they were random in order to understand properties of the original system. Examples of deterministic sequences of numbers that can be approached this way arise in problems of number theory and quantum mechanics; these provide the focus of this project. In these settings, it is possible to phrase the questions addressed by this project using the language of probability theory, but the methods required are of a different nature and vary according to the application, coming from the theory of dynamical systems. The PI has already successfully achieved several results on these topics, and has been active in the fast-advancing interplay between dynamics and number theory. The PI plans to study the statistical properties of deterministic processes of number theoretical origin such as Weyl exponential sums, multiplicative arithmetic functions, and continued fraction expansions. Although several properties of processes arising from dynamical systems are known, the PI aims to investigate finer details and answer novel questions in the field. It should be stressed that sometimes the limiting distributions agree with the classical ones of probability theory (which means that the deterministic phenomenon of interest is a good stochastic proxy); on the other hand, often one gets non-standard distributions. This means that the classical probabilistic methods cannot be applied and the theory requires new ideas and tools. The main methods in this project are of dynamical nature, such as equidistribution of one-parameter flows in homogeneous spaces, Ratner theory, and spectral methods. The PI has already successfully achieved several results on the topics, and has been active in the fast-advancing interplay between dynamics and number theory. The multifaceted aspects of this proposal will serve as an opportunity for the PI to initiate activities (such as interactive seminars, symposia, workshops) with the aim of popularizing the research subjects and their applications.