Ergodic and Geometric Properties of Diffeomorphisms

点击翻译按钮获取中文摘要

Wilkinson will continue to extend her previous work toward an understandingof the dynamics of partially hyperbolic diffeomorphisms.In particular, with her collaborators, Wilkinson proposes to addressthe 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 pointsin a space. A diffeomorphism is a natural object, arising, forexample, in the study of planetary motion. By applying a diffeomorphism repeatedly, one obtains a dynamical system; each iterationcorresponds to a unit of time in the evolution of the system. For any given space, there are infinitely many diffeomorphisms, with widelyvarying 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-understoodare Smale's Axiom A, or hyperbolic, diffeomorphisms, which are chaotic: among other things, they display sensitivedependence on initial conditions. This project aims to betterunderstand 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.