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.
Anosov dynamical systems are prime examples of dynamical systems that exhibit behaviour known as deterministic chaos. The latter term refers to the fact that the future behaviour of the dynamical system is completely dermined by its present state yet an observer sees chaotic evolution due to initial error in measurement and exponential divergence of trajectories of nearby states. The project was concerned with the space of Anosov dynamical systems, that is, the collection of all Anosov dynamical systems on given space. The following are some of the findings of the project. In collaboration with F.T. Farrell, new ``exotic" examples of Anosov dynamical systems (and their relatives expanding maps) were found. In collaboration with F.T. Farrell, the PI showed that the space of higher dimensional Anosov diffeomorphisms is a very intricate space, i.e. has many connected components and non-zero higher homotopy groups. In collaboration with F. Rodriguez Hertz, the PI showed that certain kinds of spaces, such as products of even dimensional spheres, do not support any Anosov diffeomorphisms. In collaboration with B. Kalinin and V. Sadovskaya, a local rigidity result for a large classe of Anosov automorphisms of tori was estblished. This result provides insight into the local structure of the space of Anosov dynamical systems. The broder impact activities include the following. The PI lectured on dynamical systems at the undergraduate Math Club at SUNY Binghamton. In the Fall 2011, 2012 and 2013 the PI gave an introductory course at SUNY Binghamton on dynamical systems and their applications to science and engineering. The PI guided and encouraged highly motivated students to learn advanced material that went beyond the syllabus of the course. The PI has initiated and organized video-recordings of Dean’s Lecture Series by distinguished visitors at SUNY Binghamton. The PI is/was the co-organizer of several seminars at SUNY Binghamton: Colloquium, Analysis Seminar, Geometry-Topology Seminar and Toeholds on Topology.
