Limit Theorems in Dynamical Systems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

It is well known that many deterministic systems exhibit random behavior. The simplest example of one such system is provided by a ball moving between two infinitely heavy periodically moving walls. This example was studied by Fermi and Ulam in the middle of the last century. More complicated examples include the motion of charged particles in plasma and the motion of the asteroids in the asteroid belt. The aim of this proposal is to understand the origin of stochasticity (randomness of motion) of such systems using recent advances in chaotic dynamics, with the goals of making precise predictions about their behavior. The PI will continue his work on limit theorems for dynamical systems combing the methods of probability theory and dynamical systems. The main directions of research will be the following: (1) Evolution of adiabatic invariants, and (2) Local Limit Theorems. This proposal will concentrate on the methods of obtaining precise asymptotics which allows control of the limit distributions on long temporal scales and short spatial scales. The broader impact of the proposed activity will be in applying dynamical systems tools to other fields including probability theory, number theory, and statistical physics. The PI will continue his synergistic activities including organizing conferences, giving lectures and minicourses on recent advances in dynamics, writing survey papers and mentoring graduate students.