Dynamics of Partially Hyperbolic Systems

部分双曲系统的动力学

• 批准号: 0100416
• 负责人: Keith Burns
• 金额: $ 8.55万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100416&HistoricalAwards=false
• 关键词: Dynamics; Partially; Hyperbolic; Systems

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.

该项目将研究部分双曲线系统的动态。希望通过削弱thecenter束假设并适应证据，以改善PUGH和SHUB的最新定理，以便它适用于Pointis（或巴西）的部分均匀性，而不是偏向均匀的均匀性。 Pugh和Shub做出的假设。我还希望扩展部分双曲线图的类别，其中众所周知，通过研究我本人和维尔金森已经研究的紧凑型组扩展的紧凑型组扩展，可以通过研究pogh-shub定理的假设。此外，我计划继续与Paternain onmagnetic流量进行合作，并与Hasselblatt和Wilkinson合作，以在Lyapunov指数的敏捷中进行地理流量。系统的可能状态和themanifold的可区分映射到本身，这代表了系统从其电流状态到下一个状态的演变。一种倾向于产生回教行为的基本机制是对地图的衍生物伸展向量的衍生物，并在互补方向上缩小向量。如果除了扩展和收缩方向，该系统被称为部分流血，并且缩小了第三个方向，它比扩展的方向伸展，而缩小的方向则比缩小的方向比收缩更小。混乱的行为是完全夸张的系统。部分双曲线系统的许多特定示例。