Geometry and Statistics of Measure Preserving Dynamical Systems: Theory and Applications

Igor Mezic, AM, Caltech

A geometric theory for the analysis of chaotic motion and transport in a class of 3-D incompressible fluid flows with symmetry is presented. The ingredients are the action-angle-angle variables, 3-D Melnikov theory, and 3-D KAM-type theory. Moreover, we present a study of statistical properties of dynamical systems using Birkhoff's Ergodic Theorem, ergodic partition, and methods of probability theory. We show that, in the case when the system is not ergodic, the only quantities necessary to describe the limiting (when the time or the number of iterations $\rightarrow \infty$) behaviour of these systems are the time averages. Using this observation, we derive neccessary and sufficient condition for the ubiquitous $t^2$ asymptotic behavior of the dispersion. We obtain a link between probability distributions of sum functions and the ergodic partition, and we analyze the problem of first passage times, proving some conjectures motivated by numerical experiments. The theory is developed for both maps and flows, and has applications in a variety of problems related to the statistical description of chaotic motion in physical systems.