Partial hyperbolicity and rigidity

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This project is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed over the past forty years. In particular, the principal investigator will study the following topics: ergodic properties of conservative partially hyperbolic diffeomorphisms; physical measures for (dissipative) partially hyperbolic diffeomorphisms; rigidity phenomena connected to partially hyperbolic group actions; and ergodicty of singular partially hyperbolic systems. These research goals will be carried out through a variety of modalities, including published papers in peer-reviewed journals, a collaborative book project, supervising Ph.D. students, and speaking at conferences, including the upcoming International Congress of Mathematicians in Hyderabad, India.The principal investigator is a leading expert in the field of partially hyperbolic dynamics. A dynamical system is a system (for example, a state space for a physical process) that evolves over time according to a deterministic set of rules. Well-studied classes of dynamical systems include the so-called hyperbolic systems, which display chaotic, unpredictable features at every point, and KAM systems, which have stable regions of regular motion. The partially hyperbolic systems are a more general class of dynamical systems than the hyperbolic class, and include systems that combine hyperbolicity in some directions with KAM behavior in other directions. Partially hyperbolic systems occur widely in dynamical systems that arise in physics. For example, planetary motion usually contains partially hyperbolic subdynamics. The principal investigator has had a well-developed research plan for over fifteen years to study partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability.