Rigidity in Rank One Hyperbolic Dynamics and Related Topics

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This research project is in the field of hyperbolic dynamical systems. This field encompasses a variety of real-world dynamical systems, or systems that change with time. Examples could be as simple as the motion of a mechanical linkage in the plane or in 3-dimensional space, and as complex as the behavior of gas in a closed vessel or the dynamics of convection currents in the atmosphere. Hyperbolic dynamics is colloquially known as “chaos theory” and studies chaotic (or “unpredictability” properties) and long-term ``behavior on the average” of such dynamical systems. The project seeks to make significant progress on certain foundational questions in hyperbolic dynamics. Potentially this research could have impact in physics, chemistry, engineering, and other areas where chaotic dynamical systems naturally appear. The project also has an educational component and supports the training of graduate students.Anosov diffeomorphisms, partially hyperbolic diffeomorphisms and geodesic flows in negative curvature are the prime examples of chaotic dynamical systems. In the past decades, the study of statistical and geometric properties of individual dynamical systems has flourished. However, we still have a rather poor understanding of the structure of the space of Anosov and partially hyperbolic diffeomorphisms and flows. The project seeks to gain such understanding primarily focusing on the structure of smooth conjugacy classes. The principal investigator plans to utilize an arsenal of dynamics, analytic and geometric techniques to achieve the goals of this proposal. More specifically the project has three major parts: (1) gaining fundamental understanding of the structure of smooth conjugacy classes of higher dimensional Anosov diffeomorphisms and 3-dimensional Anosov flows; (2) advancing the rigidity program for negatively curved surfaces and higher dimensional manifolds; (3) advancing the rigidity program from hyperbolic to partially hyperbolic dynamics.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.