The proposed work is concerned with the space of Anosov diffeomorphisms of a compact manifold. Anosov systems constitute the paradigm of chaotic dynamics and have been intensively studied since the late sixties. Still, some questions are wide open, most importantly, the question of classification of Anosov diffeomorphisms. There are three major directions in the proposed program. 1. Smooth classification of Anosov systems. This will continue the research program started in the PI's recent work and aims at gaining fundamental understanding of the structure of the space of higher dimensional Anosov systems. 2. Study of the global topology of the space of Anosov diffeomorphisms and the Teichmuller space of smooth conjugacy classes. 3. Anosov diffeomorphisms on simply connected four dimensional manifolds. Dimension four is the least dimension where the classification problem is open. It is expected that simply connected 4-manifolds do not support Anosov diffeomorphisms.
The project lies in the area of hyperbolic dynamics and aims at a fine understanding of hyperbolic dynamical systems. A dynamical system is called hyperbolic if the ambient space exhibits a lot of stretching in one direction and contraction in another direction. The behavior one observes in hyperbolic systems is usually referred to as deterministic chaos. One of the first appearances of such systems was in the work of Cartwright and Littlewood who studied certain differential equations of second order with periodic forcing. This work arose from war-related studies involving radio waves. Since then hyperbolic dynamics has been an important tool in natural sciences. The results of the proposed research will be disseminated through seminar and conference talks, online preprint servers as well as journal publications. Also, the PI plans to deliver expository talks in his area with the purpose of introducing undergraduate and graduate students to the exciting area of hyperbolic dynamics.
