Cohomology, Skew-Products, and Partially Hyperbolic Diffeomorphisms

这项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，其中一个有趣的问题是将Livsic的上同调结果推广到周期数据包含在李群的半群中的上循环。在这个方向上的积极结果将是前人工作的自然推广，并将给双曲作用上扩张的拓扑传递性带来障碍。第二个目标是将上同调结果用作部分双曲高阶晶格作用的刚性理论中的工具。这个目标是由Zimmer发起的刚性程序的一部分，目的是对紧致流形上保持较高秩格作用的体积进行分类。一个可达的目标是将SL(n，Z)完全非辛作用的小GL(n，R)扩张分类(即具有稳定和不稳定叶的特殊结构的双曲作用)。第三个目标是寻找具有丰富动力性质的部分双曲变换的一般类。Pugh和Shub最近猜想，在可微映射中，稳定的遍历性和可达性比预期的出现得更频繁，并且某些部分双曲性足以证明稳定的遍历性。根据Nitica和Torok的结果，得出在技术条件下，具有一维中心叶理的部分双曲微分同胚类的稳定遍历性是开的和稠密的。即使在较高的正则性类中，密度也是成立的。一个有趣的问题是将这一结果推广到更一般的部分双曲微分同胚类。另一方面，Nitica和Pollicott最近的一个结果表明，对于次流形上Anosov微分同胚的欧氏扩张，稳定传递性的唯一障碍是上同调性质的。人们希望将这一结果推广到非阿贝尔纤维，并对影响拓扑传递性的障碍进行分类。研究人员(与I.Melbourne和A.Torok)在这个方向上的最新结果证明了具有Sp(N)纤维的稳定传递扩张的存在性。混沌映射具有密集的周期点集，即被映射的高次迭代固定的点，以及传递点，对于这些点，高次迭代将任意接近于任何其他点。混乱的行为预计将在大类地图和自然界中普遍存在。这项研究的一部分将集中在寻找新的机制来产生大类动力系统的混沌，这些动力系统只表现出部分双曲性。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的兴趣，这些科学界参与了非线性动力学在物理科学中的应用。这项工作的一个副产品将是对特殊欧几里德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学中具有有趣的应用。该调查员积极参与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。这些活动将从赠款中受益。

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.