Cocycles play an important role in classical dynamics as well as in the theory of group actions. The main part of the proposed research is the study of cocycles over hyperbolic and partially hyperbolic diffeomorphisms. In this context important examples of linear cocycles are given by the differential of the diffeomorphism and by its restrictions to invariant sub-bundles of the tangent bundle. If the bundle is trivial the linear cocycle can be viewed as a cocycle with values in a matrix group. The PI will also study cocycles with values in other groups, including infinite dimensional ones. The PI plans to work on various problems in cohomology of cocycles. The questions in this area are motivated in part by problems in smooth dynamics and rigidity of hyperbolic and partially hyperbolic systems and group actions. The PI intends to use her recent results as well as the projected outcomes to further the research in these areas.
Dynamics is the area of mathematics that studies evolution of a system over time with an emphasis on describing its long term behavior. The systems under consideration come from mathematics, physics, and other sciences. Hyperbolic and partially hyperbolic systems have been one of the main objects of study in differentiable dynamics. Exponential contraction and expansion in these systems produce chaotic behavior, and understanding the complex structure of the system is an important goal. The derivative plays a crucial role in analysing these systems, and it gives an example of a cocycle over the system. Cocycles also appear naturally in linearized equations that relate two systems and thus are useful in the study of stability, rigidity, and classification of hyperbolic systems.
