Non-Commutative Cocycles and Dynamics of Systems with Hyperbolic Behavior

The theory of dynamical systems is a modern branch of mathematics which originated from the study of physical and mechanical problems. It describes how various abstract and real-life systems evolve over time, and so it has a wide range of applicability. This project is focused on systems that exhibit hyperbolic behavior, that is, exponential expansion in some directions and exponential contraction in other directions. The expansion and contraction produce a rich and complex behavior of the system, often described as chaotic, with individual trajectories highly sensitive to small changes in the initial conditions. Nonetheless, if a system is hyperbolic in a strong sense it is stable as a whole, that is, qualitatively similar to any small perturbation. Cocycles are a fundamental tool in the study of dynamical systems. For systems given by differentiable functions, the derivative and related objects are the prime examples of cocycles. Another important class of examples is given by random sequences of matrices or maps. Cocycles are useful in studying when two dynamical systems are similar and to what extent and, in particular, when a system is similar to a perturbation or to a standard model.

Cocycles appear naturally in dynamics and, more generally, in group actions. The principal investigator will study cocycles over hyperbolic, partially hyperbolic, and non-uniformly hyperbolic dynamical systems. The research will be focused on cocycles with values in non-commutative non-compact groups, such as the general linear group, groups of linear operators on Hilbert and Banach spaces, groups of diffeomorphisms of compact manifolds, and groups of isometries of spaces of non-positive curvature and their generalizations. The principal investigator will investigate cohomology of cocycles, regularity of a conjugacy between two cocycles, and existence of conjugacy to simpler cocycles. The principal investigator will also work on related problems of estimating growth of a cocycle and its Lyapunov exponents, with a focus on using the periodic data. These questions are motivated in part by problems in smooth dynamics and rigidity of systems and actions exhibiting some hyperbolicity. The principal investigator will use the results on cohomology of cocycles and on non-stationary normal forms to advance the development of this area. The principal investigator will focus on topological and smooth rigidity of a single hyperbolic system and on global rigidity of higher rank hyperbolic abelian actions on an arbitrary manifold.

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.