Beyond Renormalization in Parabolic Dynamics

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The theory of dynamical systems studies the long term behavior of deterministic systems. A classical example is the motion of the planets. Dynamical systems arise in all areas of sciences, for instance in physics, biology, and economics. In addition, the methods of dynamical systems can be applied to study problems in other fields of mathematics, including geometry and number theory. There is a well-established theory of chaos that applies to dynamical systems whose nearby trajectories diverge exponentially rapidly with time (the weather is perhaps the most famous example). At the other end of the spectrum, there is regular motion, characterized by slowly-evolving families of trajectories. This research project aims to advance fundamental research on the intermediate case of weakly chaotic systems, often called parabolic. Parabolic systems are characterized by the property that nearby trajectories diverge at most polynomially rapidly with time. Systems of this kind are especially important in geometry, number theory, and mathematical models arising in several branches of physics, including solid-state physics, celestial mechanics, and statistical mechanics. For instance, in recent years, in number theory, and to a lesser extent, in geometry, many questions have been reformulated (sometimes solved) as questions on the dynamics of certain parabolic flows. In physics, the behavior of electrons at the Fermi surface (solid-state physics), the motion of planets near a singularity (celestial mechanics), and the motion of a confined atom (statistical mechanics) are related to parabolic systems. This project aims to develop new approaches to these important questions and will involve graduate students in the fundamental mathematical research. Dynamical systems can be roughly distinguished according to the speed of divergence of nearby trajectories. Systems with sub-exponential, polynomial divergence of nearby orbits are often called parabolic. An extremely successful approach to some parabolic systems is based on renormalization, which is an effective tool whenever the system reveals itself to be at least approximately self-similar, meaning it is nearly the same when viewed at smaller and smaller scales, possibly after changing of coordinates. However, many fundamental parabolic systems, including most homogeneous flows, do not seem to possess self-similarity properties, and no renormalization method has been developed for such systems. As a consequence, the dynamical properties of these systems are very poorly understood and are rarely the subject of investigation. This project aims to build on recent results by the principal investigator and collaborators to investigate non-renormalizable systems. The guiding principle is to develop a scaling method to generalize renormalization in the absence of self-similarity. Non-renormalizable parabolic systems include higher-step nilflows and higher rank Abelian actions on higher step nilmanifolds. This includes most unipotent Abelian actions on quotients of semisimple Lie groups and billiards in non-rational polygons.