Groups and Hyperbolic Dynamical Systems

Texas A&M University, College Station TX

Investigators

Abstract

The project explores algebraic aspects of chaotic dynamical systems, namely systems with hyperbolic behavior. This is a classical type of system, which has its historical roots in the study of the famous three-body problem of physics. Even after decades of investigations, many basic questions about such systems remain open. For example, it is not known which of these systems have a rigidly algebraic description (can be constructed from groups). The project aims at studying the connections of hyperbolic dynamical systems with group theory, the theory of automata, and other branches of algebra. Conversely, the project plans to use the insights gained from the study of hyperbolic systems to solve open questions in group theory. The project also includes training and advising of students, organization of an annual high school mathematics competition, and the completion of a graduate textbook on groups and topological dynamics. Hyperbolic dynamical systems are classical and well-studied examples of chaotic dynamics. However, several basic open questions remain open. One such question is whether every hyperbolic homeomorphism of a locally connected compact space comes from a nilpotent Lie group. The question is probably very hard, but the project outlines several directions toward clarifying group-theoretic aspects. The project aims to study automatic structures associated with hyperbolic systems and groups generated by such automatic structures. These groups may have interesting applications to the theory of the growth of groups and amenability. The project's plans also include the study of the relation between group theory (contraction coefficients of virtual endomorphisms) and the conformal dimension of hyperbolic dynamical systems, which can have interesting applications of group theory to rigidity results in dynamics and to asymptotic group theory. 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.