Limit Theorems in Dynamical Systems

This project develops methods to carry out precise computations of statistical quantities associated with chaotic behavior of deterministic dynamical systems. Deterministic systems with unstable behavior appear in a wide range of applications, ranging from atomic to planetary scales. In many cases the instability is caused by exponential divergence of nearby trajectories. Long time behavior of such systems cannot be predicted exactly unless the initial conditions are known precisely, and consequently it is necessary to use statistical descriptions of the orbits. The project also addresses simple models of mixed dynamical behavior where systems exhibit features associated with both regular and stochastic behaviors. Such models occur as paradigmatic examples explaining more complicated phenomena. In addition, the project provides opportunities for research training and mentoring of graduate students. The principal investigator will also disseminate results through lectures and expository writing. The project focuses on the statistical properties of dynamical systems. It consists of four parts. First, precise asymptotics in limit theorems for dynamical systems will be considered. Sharp limit theorem estimates have recently been shown by the principal investigator and others to have applications in dynamics. A local limit theorem for non-autonomous systems will be studied. The principal investigator plans to classify obstructions to classical asymptotic expansions, and to develop new tools applicable in cases where the classical expansions fail. A second topic to be considered is exponential mixing. Sharp limit theorems for exponentially mixing systems will be studied through an exploration of geometric structures in the associated phase space. Next, the principal investigator will study flexibility of statistical properties through the construction of dynamical systems with exotic properties. A final topic under consideration is averaging theory. The principal investigator has previously made several contributions to the study of averaging for systems with hyperbolic fast motion. These techniques will be investigated further with an eye towards extensions to the setting of quasi-periodic fast dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.