New Methods for Smooth Rigidity of Algebraic Actions

Michigan State University, East Lansing MI

Investigators

Abstract

The field of dynamical systems originated from differential equations and celestial mechanics. It studies the long run behavior of a system subject to laws of motion. So far, systems with strong chaotic properties have been well understood. These results provide applications to other areas of mathematics as well as to many areas of sciences such as physics, mechanics, computer science, and biology. In many systems of interest however, only weak chaotic behaviors can be observed. The goal of this project is to develop new tools to study such systems; and then apply these results to study other areas of mathematics, such as number theory and representation theory. There is a growing interest in recent years for irreducible higher rank algebraic actions. Many examples of these actions exhibit a remarkable array of rigidity properties. These examples possess strong chaotic properties. This project aims at providing rigidity examples with less chaotic behaviors. This project will study smooth rigidity of a broad class of algebraic actions, especially for the actions completely absent of hyperbolicity, like parabolic actions. Current tools fail to be applied to treat these actions. The main research theme in this project is to develop a method that combines classical KAM (Kolmogorov-Arnold-Moser) approach with representation theory to study the rigidity behavior of a broad class of algebraic actions. The new method will afford the first local rigidity examples for parabolic actions; and has the potential to establish local rigidity for partially hyperbolic actions whose geometric properties are distinctly different from existing examples.