In this project the principal investigator will be continue his studies in dynamics with a particular emphasis on partially hyperbolic dynamical systems. Partial hyperbolicity has seen great progress in the last decade after the breakthrough work of Pugh and Shub, who established ergodicity for volume-preserving perturbations of the time-one map of the geodesic flow of a surface of constant negative curvature. A series of papers, culminating in work of the principal investigator with Wilkinson, has established ergodicity for a very general class of volume-preserving partially hyperbolic systems. These results appear to have pushed current techniques to their limit, but still fall short of providing a complete understanding of when volume-preserving partially hyperbolic systems are ergodic: new ideas are needed. This project will pursue such fresh lines of attack on this question. It will also consider various dynamical systems of a geometrical nature that exhibit hyperbolic behavior.
Partially hyperbolic dynamical systems are important because they model significant phenomena that occur in nature and because the mathematical tools for understanding such systems are by now well developed. For example, there is now a good understanding of how partial hyperbolicity gives rise to highly chaotic behavior. Researchers also possess a rich set of concrete examples in which it is possible to understand completely the mechanisms that create the partially hyperbolic behavior. In the long run, improving this knowledge, as this project hopes to do, could have important implications for science and engineering.
