Burns will be continuing his studies of partially hyperbolic dynamical systems. This area 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 recent work of Burns and Wilkinson, has established ergodicity for a very general class of volume preserving partially hyperbolic systems. This work appears to have pushed current techniques to their limit, but still falls short of a complete understanding of when volume preserving partially hyperbolic systems are ergodic; new ideas are needed. Burns will continue to study this question as well as various special examples of partially hyperbolic dynamical systems, in particular compact group extensions of hyperbolic basic sets. He will also consider geometrical questions about the geodesic which are closely related to the work on partial hyperbolicity.
Partially hyperbolic dynamical systems are important because they model phenomena which occur in nature and because, at least in some examples, it is possible to completely understand the mechanisms which create the observed phenomena. Partial hyperbolicity is to be expected in systems which have a fast-slow dynamics in which one aspect of the systems operates on a short time scale and other aspects operate on a longer time scale. There is now a good understanding of how partial hyperbolicity gives rise to highly chaotic behaviour in many situations.
