Ergodic and Geometric Properties of Diffeomorphisms

Wilkinson将继续扩展她以前的工作，以了解部分双曲微分同态的动力学。特别是，Wilkinson和她的合作者建议解决部分双曲微分同态空间中稳定遍历性的密度问题，以及这些微分同态的Sinai-Ruelle-Bowen测度的存在性。Wilkinson还将研究部分双曲系统的不变结构对光滑参数的依赖性。微分同态是一种光滑地重新排列空间中的点的方法。微分同胚是一种自然的物体，例如，在研究行星运动时出现的。通过重复应用微分同胚，我们得到了一个动力系统；每次迭代对应于系统演化中的一个时间单位。对于任何给定的空间，都有无限多的微分同胚，它们的动力学行为有很大的不同；动力学的一个目标是将微分同胚安排成其动力学行为可以理解的类。一类动力学现在被很好地理解的微分同胚是斯梅尔公理A，或双曲微分同胚，它们是混沌的：在其他方面，它们表现出对初始条件的敏感依赖。这个项目旨在更好地理解另一类重要的微分同胚的行为，称为部分双曲微分同胚。部分双曲微分同胚具有公理A系统的许多特征，但其动力学行为尚不完全清楚。

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.