Cohomology, Skew-Products, and Partially Hyperbolic Diffeomorphisms

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，其中一个目标是将Livsic的上同调结果推广到李群中的半群中包含周期数据的上循环。这方面的结果是对前人工作的自然推广，同时也给双曲作用上扩张的拓扑传递性带来了障碍。第二个目标是利用上同调结果作为部分双曲高阶格点作用量刚性理论的工具。这一目标是由Zimmer发起的刚性计划的一部分，旨在将紧致流形上的高秩格作用量分类。一个可达到的目标是分类SL（n，Z）的完全非辛作用（即双曲的，具有稳定和不稳定叶理的特殊结构）的小GL（n，R）扩张。第三个目标是寻找具有丰富动力学性质的部分双曲变换的一般类。Pugh和Shub最近指出，在可微映射中，稳定遍历性和可达性比预期的更频繁地出现，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型同态类的稳定遍历性是开的和稠密的。密度甚至在更高的正则类中也成立。一个有趣的问题是把这个结果推广到更一般的部分双曲型双同态类。在另一个方向上，Nitica和Pollicott最近的一个结果表明，对于infranilmanifold上的Anosov同态的欧氏扩张，唯一的稳定传递性障碍是上同调性质。人们希望将这一结果推广到非阿贝尔纤维，并对拓扑传递性的障碍进行分类。调查员（I。墨尔本和A. Torok）在这方面的最新结果证明了存在Sp（n）纤维的稳定传递扩张。混沌映射有一个稠密的周期点集，即由映射的更高阶数固定的点，以及传递点，即更高阶数将任意接近任何其他点的点。混沌行为在大类映射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新的机制，产生混乱的大类动力系统，只表现出部分双曲。它还将集中精力寻找混乱行为的障碍。这些结果将是感兴趣的更广泛的科学界参与在物理科学中的非线性动力学的应用。这项工作的副产品将是一个仔细研究的半群在特殊的欧几里德群体与有趣的应用离散控制理论和机器人。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。这些活动将受益于赠款。

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.