Measure rigidity and smooth rigidity of abelian actions

The proposed research is in the realm of dynamical systems and smooth ergodic theory. The main goal of the project is to study smooth actions of (discrete and continuous) higher rank Abelian groups, actions that exhibit some degree of hyperbolicity. They represent natural generalizations of diffeomorphisms and flows on smooth manifolds. Such higher rank dynamical systems appear in various geometric and algebraic situations. The main examples include actions by commuting hyperbolic and partially hyperbolic automorphisms of tori and by commuting translations on cosets of Lie groups. In contrast to the properties of a single hyperbolic diffeomorphism or flow, many remarkable rigidity phenomena have been discovered for certain classes of higher rank Abelian actions. These phenomena include smooth rigidity, cocycle rigidity, and rigidity of invariant measures. Using dynamical, analytic, and group theoretic methods the principal investigator will further the study of such rigidity properties. In particular, he will consider broader classes of these actions than have been investigated hitherto, including nonalgebraic and nonuniformly hyperbolic actions. Another goal of the project is to study certain questions in smooth dynamics of a single Anosov or partially hyperbolic system.

Dynamical systems and ergodic theory are relatively new fields of mathematics that study the evolution of physical and mathematical systems over time (e.g., planetary motion, flow of air or other fluids). With origins in differential equations and celestial mechanics, these modern branches of mathematics provide numerous applications not only to other areas of mathematics but also to such natural sciences as physics, biology, meteorology, sociology, and computer science. Dynamical systems and ergodic theory have introduced new mathematical ideas into these sciences, including the study of long-term qualitative behavior, along with various analytic and probabilistic methods. One of the main goals of the proposed research is to investigate complex dynamical systems consisting of several different systems that ?commute? with each other, meaning that the evolution of any of the systems is not affected by the changes brought about by the others. Complex systems of this kind appear naturally in algebra, geometry, and physics. The recent study of such systems has produced exciting applications to number theory and quantum mechanics.