Wilkinson will continue to extend her previous work toward an understanding of the dynamics of partially hyperbolic diffeomorphisms. In particular, with her collaborators, Wilkinson proposes to address the issue of density of stable ergodicity in the space of partially hyperbolic diffeomorphisms, and the existence of Sinai-Ruelle-Bowen measures for these diffeomorphisms. Wilkinson will also study the dependence on smooth parameter of invariant structures for partially hyperbolic systems.
A diffeomorphism is a way of smoothly rearranging the points in a space. A diffeomorphism is a natural object, arising, for example, in the study of planetary motion. By applying a diffeomorphism repeatedly, one obtains a dynamical system; each iteration corresponds to a unit of time in the evolution of the system. For any given space, there are infinitely many diffeomorphisms, with widely varying dynamical behaviors; a goal of dynamics is to arrange diffeomorphisms into classes whose dynamical behavior can be understood. One class of diffeomorphisms whose dynamics are now well-understood are Smale's Axiom A, or hyperbolic, diffeomorphisms, which are chaotic: among other things, they display sensitive dependence on initial conditions. This project aims to better understand the behavior of another important class of diffeomorphisms, called partially hyperbolic diffeomorphisms. Partially hyperbolic diffeomorphisms have many of the features of Axiom A systems, but their dynamical behavior is yet to be completely understood.