Non-Commutative Cocycles and Dynamics of Systems with Hyperbolic Behavior
非交换余循环和具有双曲行为的系统动力学
基本信息
- 批准号:1764216
- 负责人:
- 金额:$ 12.57万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2018
- 资助国家:美国
- 起止时间:2018-06-15 至 2022-05-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The theory of dynamical systems is a modern branch of mathematics which originated from the study of physical and mechanical problems. It describes how various abstract and real-life systems evolve over time, and so it has a wide range of applicability. This project is focused on systems that exhibit hyperbolic behavior, that is, exponential expansion in some directions and exponential contraction in other directions. The expansion and contraction produce a rich and complex behavior of the system, often described as chaotic, with individual trajectories highly sensitive to small changes in the initial conditions. Nonetheless, if a system is hyperbolic in a strong sense it is stable as a whole, that is, qualitatively similar to any small perturbation. Cocycles are a fundamental tool in the study of dynamical systems. For systems given by differentiable functions, the derivative and related objects are the prime examples of cocycles. Another important class of examples is given by random sequences of matrices or maps. Cocycles are useful in studying when two dynamical systems are similar and to what extent and, in particular, when a system is similar to a perturbation or to a standard model.Cocycles appear naturally in dynamics and, more generally, in group actions. The principal investigator will study cocycles over hyperbolic, partially hyperbolic, and non-uniformly hyperbolic dynamical systems. The research will be focused on cocycles with values in non-commutative non-compact groups, such as the general linear group, groups of linear operators on Hilbert and Banach spaces, groups of diffeomorphisms of compact manifolds, and groups of isometries of spaces of non-positive curvature and their generalizations. The principal investigator will investigate cohomology of cocycles, regularity of a conjugacy between two cocycles, and existence of conjugacy to simpler cocycles. The principal investigator will also work on related problems of estimating growth of a cocycle and its Lyapunov exponents, with a focus on using the periodic data. These questions are motivated in part by problems in smooth dynamics and rigidity of systems and actions exhibiting some hyperbolicity. The principal investigator will use the results on cohomology of cocycles and on non-stationary normal forms to advance the development of this area. The principal investigator will focus on topological and smooth rigidity of a single hyperbolic system and on global rigidity of higher rank hyperbolic abelian actions on an arbitrary manifold.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
动力系统理论是数学的一个现代分支,起源于对物理力学问题的研究。它描述了各种抽象和现实生活中的系统如何随着时间的推移而演变,因此它具有广泛的适用性。这个项目的重点是系统表现出双曲线行为,即在某些方向上的指数膨胀和在其他方向上的指数收缩。膨胀和收缩产生了系统的丰富而复杂的行为,通常被描述为混沌,单个轨迹对初始条件的微小变化非常敏感。尽管如此,如果一个系统在强意义上是双曲的,它作为一个整体是稳定的,也就是说,定性地类似于任何小扰动。余圈是研究动力系统的基本工具。对于由可微函数给出的系统,导数和相关对象是上循环的主要例子。另一类重要的例子是随机序列的矩阵或映射。上循环在研究两个动力学系统相似的程度,特别是当一个系统类似于一个扰动或一个标准模型时是很有用的。上循环自然地出现在动力学中,更一般地,在群作用中。主要研究者将研究双曲,部分双曲和非一致双曲动力系统上的余循环。研究将集中在非交换非紧群中的值的上循环,例如一般线性群,Hilbert和Banach空间上的线性算子群,紧流形的非同构群,以及非正曲率空间的等距群及其推广。主要研究者将研究上循环的上同调,两个上循环之间共轭的正则性,以及更简单的上循环的共轭的存在性。首席研究员还将研究估计上循环增长及其李雅普诺夫指数的相关问题,重点是使用周期数据。这些问题的动机部分问题的顺利动态和刚性的系统和行动表现出一定的双曲性。主要研究者将使用上同调的上循环和非平稳正常形式的结果,以推进这一领域的发展。首席研究员将专注于拓扑和光滑刚性的一个单一的双曲系统和全球刚性的高阶双曲阿贝尔行动的任意manifold.This奖项反映了NSF的法定使命,并已被认为是值得的支持,通过评估使用基金会的智力价值和更广泛的影响审查标准。
项目成果
期刊论文数量(2)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems
双曲系统上微分同胚余循环的有界性和不变度量
- DOI:10.1007/s10711-019-00421-9
- 发表时间:2019
- 期刊:
- 影响因子:0.5
- 作者:Sadovskaya, Victoria
- 通讯作者:Sadovskaya, Victoria
Local rigidity of Lyapunov spectrum for toral automorphisms
环自同构的李亚普诺夫谱的局部刚性
- DOI:10.1007/s11856-020-2028-6
- 发表时间:2020
- 期刊:
- 影响因子:1
- 作者:Gogolev, Andrey;Kalinin, Boris;Sadovskaya, Victoria
- 通讯作者:Sadovskaya, Victoria
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Victoria Sadovskaya其他文献
Victoria Sadovskaya的其他文献
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{{ truncateString('Victoria Sadovskaya', 18)}}的其他基金
Cocycles over hyperbolic and partially hyperbolic systems
双曲和部分双曲系统上的余循环
- 批准号:
1301693 - 财政年份:2013
- 资助金额:
$ 12.57万 - 项目类别:
Standard Grant
RUI: Invariant geometric structures and rigidity in hyperbolic dynamics.
RUI:双曲动力学中的不变几何结构和刚性。
- 批准号:
0901842 - 财政年份:2009
- 资助金额:
$ 12.57万 - 项目类别:
Standard Grant
Conformal Structures and Rigidity Properties of Anosov and Partially Hyperbolic Systems
Anosov 和部分双曲系统的共形结构和刚度性质
- 批准号:
0401014 - 财政年份:2004
- 资助金额:
$ 12.57万 - 项目类别:
Standard Grant
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