The proposed project is a broad program that aims at advancing our understanding of the structure of Anosov and partially hyperbolic systems. These dynamical systems are the prime examples of chaotic dynamical systems and, by now, we have good understanding of their statistical properties. On the other hand the topology of the underlying spaces of hyperbolic dynamical systems is not well understood. For example, all known examples of Anosov diffeomorphism live on infranilmanifolds, but the outstanding classification problem of Anosov diffeomorphisms is still open. The PI proposes a wide-ranging program to approach classification questions in hyperbolic dynamics. This program includes: (1) search for higher homotopy obstructions to existence of Anosov diffeomorphism; (2) search for geometric obstructions (such as presence of negative curvature) to existence of Anosov diffeomorphism; (3) construction of expanding maps and Anosov diffeomorphisms on PL-exotic infranilmanifolds; (4) search for new examples of partially hyperbolic diffeomorphism; (5) classification of partially hyperbolic diffeomorphisms according to their action on cohomology; (6) classification of partially hyperbolic diffeomorphisms according to the induced dynamics on the space of center leaves.
Chaotic dynamical systems are abundant. The examples include: the motion of the double pendulum, free motion of elastically colliding hard balls in a rectangular box (ideal gas), atmospheric convection (the famous Lorenz attractor). The study of the examples that arise from the nature is extremely challenging and there are still many mysteries. Hyperbolic dynamical systems that the PI proposes to study are simplified mathematical models of chaotic dynamical systems that arise from the nature. Statistically the long term behavior of hyperbolic systems is relatively well understood. However, the structure of the underlying spaces for hyperbolic dynamical systems is poorly understood. The PI proposes a wide-ranging program that will advance our knowledge of spaces that support hyperbolic systems. Potentially this research may have impact in physics, biology, chemistry, engineering and other areas where chaotic dynamics arises.
