Cohomology, Skew-Products, and Partially Hyperbolic Diffeomorphisms

上同调、斜积和部分双曲微分同胚

• 批准号: 0500832
• 负责人: Viorel Nitica
• 金额: $ 8万
• 依托单位: West Chester University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500832&HistoricalAwards=false
• 关键词: Cohomology; Skew; Products; Partially; Hyperbolic

The proposed research has several goals. The first goal is tostudy cohomological equations over hyperbolic dynamical systems.One is interested to generalize Livsic's cohomologicalresults to cocycles for which periodic data is included insemigroups in Lie groups. Positive results in this direction willbe natural generalizations of previous work, and will giveobstructions to the topological transitivity of extensions overhyperbolic actions. The second goal is to use cohomologicalresults as a tool in the rigidity theory of partially hyperbolichigher rank lattice actions. This goal is part of the rigidityprogram initiated by Zimmer, aiming to classify volume preservinghigher rank lattice actions on compact manifolds. A reachabletarget is to classify small GL(n,R) extensions of SL(n,Z) totally non-symplectic actions (i.e. hyperbolic and with a special structure of thestable and unstable foliations). The third goal is to find genericclasses of partially hyperbolic transformations with rich dynamicproperties. Pugh and Shub recently conjectured that stablyergodicity and accessibility occur more frequently than expected among differentiable maps, and that some partial hyperbolicity is sufficient to prove stably ergodicity. From a result of Nitica and Torok it follows that, under technical conditions, stably ergodicity isopen and dense in the class of partially hyperbolic diffeomorphismswith one-dimensional central foliation. The density holds even in the higher regularity classes. An interesting problem is togeneralize this result to more general classes of partially hyperbolic diffeomorphisms. In other direction, a recent result of Nitica and Pollicott shows that for Euclidean extensions of Anosov diffeomorphisms on infranilmanifolds the only obstructions to stable transitivity are of cohomological nature. One would like to generalize this result to non-abelian fibers, and to classify the obstructions to topological transitivity. The Investigator (with I. Melbourne and A. Torok) has recent results in this direction proving the existence of stably transitive extensions with Sp(n) fiber.A chaotic map has a dense set of periodic points, that is points that are fixed by a higher iterate of the map, as well as transitive points, that is points for which a higher iterate will get arbitrarily close to any other point. Chaotic behaviour is expected to be generic in large classes of maps and in nature. Part of this research will be focused on finding new mechanisms for producing chaos for large classes of dynamical systems that exhibit only partial hyperbolicity. It will also concentrate on finding obstructions to chaotic behaviour. These results will be of interest to the broader scientific community involved in applications of nonlinear dynamics in physical sciences. A spin-off of this work will be a careful study of the semigroups in the Special Euclidean groups with interesting applications to discrete control theory and robotics. The Investigator actively participates in recruitment, training and professional development of K-12 mathematics teachers. He will continue his collaboration with undergraduate students and will support them to give talks at professional meetings. These activities will benefit from the grant.

拟议的研究有几个目标。第一个目标是研究双曲动力系统上的上同调方程。我们有兴趣将Livsic的上同调结果推广到周期数据包含在李群半群中的共环上。在这个方向上的积极结果将是对以往工作的自然推广，并将阻碍扩展过双曲作用的拓扑及性。第二个目标是利用上同调结果作为部分双曲高阶格作用的刚性理论的工具。这个目标是由Zimmer发起的刚性计划的一部分，旨在对紧流形上的体积保持高秩点阵行为进行分类。一个可达到的目标是对SL（n，Z）的完全非辛作用（即双曲和具有稳定和不稳定叶的特殊结构）的GL（n，R）小扩展进行分类。第三个目标是找到具有丰富动态特性的部分双曲变换的泛类。Pugh和Shub最近推测，稳定遍历性和可达性在可微映射中出现的频率比预期的要高，并且某些部分双曲性足以证明稳定遍历性。从Nitica和Torok的结果可以得出，在技术条件下，具有一维中心叶理的部分双曲微分同构类的稳定遍历性是开的和密的。即使在更高的正则类中，密度也保持不变。一个有趣的问题是将这个结果推广到更一般的部分双曲微分同态。在另一个方向上，Nitica和Pollicott最近的结果表明，对于基础流形上的Anosov微分同态的欧几里德扩展，稳定传递的唯一障碍是上同调性质。我们想把这个结果推广到非阿贝尔纤维上，并对拓扑传递性的障碍进行分类。研究者（与I. Melbourne和A. Torok）在这个方向上有最近的结果，证明了Sp(n)光纤存在稳定的可传递扩展。混沌映射有一个密集的周期点集合，这些点是由映射的高迭代固定的，还有传递点，这些点的高迭代会任意接近任何其他点。混沌行为在大型地图和自然界中是普遍存在的。这项研究的一部分将集中在寻找产生混沌的新机制，为大类别的动力系统，只显示部分双曲。它还将专注于寻找混乱行为的障碍。这些结果将引起更广泛的科学界对非线性动力学在物理科学中的应用的兴趣。这项工作的副产品将是对特殊欧几里得群中的半群的仔细研究，这些半群在离散控制理论和机器人技术中有有趣的应用。研究者积极参与K-12数学教师的招聘、培训和专业发展。他将继续与本科生合作，并支持他们在专业会议上发表演讲。这些活动将受益于赠款。