Dynamics of Partially Hyperbolic Systems

本项目将研究部分双曲系统的动力学，希望通过削弱中心聚束假设和修改证明来改进Pugh和Shub最近的定理，使其适用于部分双曲的逐点（或巴西）版本，而不是Pugh和Shub所做的更严格的一致假设。我还希望通过研究我和威尔金森已经研究过的紧群扩展的紧群扩展来扩展部分双曲映射的类，其中Pugh-Shub定理的假设是已知的。此外，我计划继续与Paternain合作研究磁流，并与Hasselblatt和威尔金森合作研究测地线流的李雅普诺夫指数，这个项目将研究部分双曲系统的动力学，可微动力系统由一个表示系统可能状态的可微流形和一个表示系统从其当前演化的流形到其自身的可微映射组成到下一个国家。产生混沌行为的一个基本机制是映射的导数在某些方向上拉伸向量而在互补方向上收缩向量，这种行为称为双曲性。如果除了伸缩的膨胀和收缩方向之外，还有第三个方向的伸缩小于膨胀方向，收缩小于收缩方向，则称该系统为部分双曲系统。长期以来，人们一直怀疑大多数部分双曲系统应该具有与完全双曲系统相同的混沌行为。在20世纪90年代，Pugh和Shub的工作（与Graysonand威尔金森合作）已经有可能证明这一点在相当大的普适性。我的目标是扩展他们的工作，通过削弱假设在其主要定理和研究一些特定的例子，部分双曲系统。

This project will investigate the dynamics of partially hyperbolic systems.It is hoped to improve the recent theorem of Pugh and Shub by weakening thecenter bunching hypothesis and adapting the proof so that it applies to thepointwise (or Brazilian) version of partial hyperbolicity rather than morestringent uniform assumptions made by Pugh and Shub. I also hope to extendthe classes of partially hyperbolic maps within which the hypotheses of thePugh-Shub theorem are known to hold generically by studying compact groupextensions of the compact group extensions already studied by myself andWilkinson. In addition I plan to continue my work with Paternain onmagnetic flows and to collaborate with Hasselblatt and Wilkinson on astudy of Lyapunov exponents for geodesic flows.This project will study the dynamics of partially hyperbolic systems.A differentiable dynamical system consists of a differentiable manifold whichrepresents the possible states of the system and a differentiable map of themanifold to itself which represents the evolution of the system from itscurrent state to its next state. A basic mechanism which tends to producechaotic behavior is for the derivative of the map to stretch vectors insome directions and to shrink vectors in the complementary directions.Such behavior is called hyperbolicity. The system is called partiallyhyperbolic if in addition to the expanding and contracting directionsthat are stretched and shrunk there is a third direction which is stretchedless than the expanding direction and shrunk less than the contracting.It has long been suspected that most partially hyperbolic systems shouldhave the same chaotic behavior as fully hyperbolic systems.In the 1990's the work of Pugh and Shub (in collaboration with Graysonand Wilkinson) has made it possible to prove this in considerable generality.I aim to extend their work, by weakening the hypotheses in their main theoremand studying a number of particular examples of partially hyperbolic systems.