Perturbations of smooth group actions and cohomology

William Marsh Rice University, Houston TX

Investigators

Abstract

The central objective of this project is to obtain a better understanding of the local structure of smooth group actions. Lack of dynamical diversity (local rigidity) among small perturbations of a group action is a rare phenomenon and is typically connected to other rigidity phenomena that have been discovered in the past two decades. It happens that some form of infinitesimal (or cohomological) rigidity may lead to local rigidity. In the realm of classical dynamical systems this was proved for Diophantine circle rotations in the 1960s and led to the KAM Theorem. This project is concerned with a general approach to describe local structure based on the cohomological data, even in the absence of cohomological rigidity. The project includes applications to several situations of interest where cohomology is not trivial but is well understood, and to situations where cohomology is simple but there is not enough analytical precision in its description. Understanding small perturbations of dynamical systems has always been one of the imperatives of science, given the fact that complete prediction of system behavior over time can typically be carried out only for mathematically simple systems. The solar system is one example of a natural system that is a small perturbation of a simple system. In this case, a celebrated mathematical theorem (the KAM Theorem, from which the current project draws its inspiration) sheds light on the stability of the system. Group actions, the focal subject of this project, can be viewed as dynamical systems with "multidimensional time," and many systems in biology (neural networks), computer science (multidimensional data storage media), and physics (quasi-crystals) can be modeled more efficiently by mathematical systems that allow for multidimensional time than by classical systems in which time is one-dimensional. This brings about the need to understand small perturbations of such systems. On the other hand, rigidity of local structure is often just one manifestation of a system that is rigid in the large. Others, including rigidity of so-called invariant measures for certain systems with multidimensional time, are closely connected to important problems in number theory.