RUI: Cocycles and rigidity for hyperbolic systems and actions

The main objects of the proposed research are smooth dynamical systems such as diffeomorphisms and flows, as well as smooth actions of higher rank abelian groups. The main goal of the project is to study various rigidity properties for hyperbolic, partially hyperbolic, and non-uniformly hyperbolic systems and actions. Higher rank actions exhibit remarkable properties including rigidity of invariant measures and smooth rigidity. In rank one case, rigidity occurs for some non-typical systems, such as those with high smoothness of invariant foliations or strong pinching. An important role in both areas is played by cocycles. Investigating cohomology of non-commutative cocycles over hyperbolic systems is a major part of the proposed research.

The field of Dynamical Systems studies evolution of mechanical, physical, and mathematical systems over time. It provides applications to other areas of mathematics as well as to many natural sciences such as physics, mechanics, computer science, and biology. This field originated from differential equations and celestial mechanics. It uses mathematical tools from analysis and topology to study qualitative behavior of systems 'in the long run'. A major part of the project is the study of more complex systems consisting of several individual ones that commute with each other. Such complex systems appear naturally in algebra, geometry, and physics. The PI works for the main regional university which is primarily undergraduate and educates most of the local school teachers.

-----------------------------------------------------