Dynamics and Ergodic Theory

动力学和遍历理论

• 批准号: 327636-2013
• 负责人: Quas, Anthony
• 金额: $ 1.38万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572079
• 关键词: Dynamics; Ergodic; Theory

The proposed research concerns a number of inter-related aspects of dynamical systems (the study of evolving processes) and ergodic theory (evolving processes with randomness). It also touches on probability theory. The principal themes are (1) the behaviour of piecewise isometries; (2) thermodynamic formalism; (3) Operator-valued Multiplicative ergodic theorems. Piecewise isometries are amongst the simplest examples of discontinuous dynamical systems. A map is defined by taking a partition of a subset of a Euclidean space and applying an isometry to each part. Due to the simplicity of the map on each partition element, whatever complexity occurs in these systems happens because of the discontinuous nature of the map. The goal here is to develop a broadly-based toolkit and theory for this class of mappings. In thermodynamic formalism, Gibbs measures are defined to be invariant measures maximizing a quantity analogous to those arising in thermodynamics. A limiting case of this is ergodic optimization in which one studies measures maximizing a potential. This corresponds thermodynamically to 'lowering the temperature to 0'. The project will study general properties of these maximizing measures. A second topic in this area is understanding the relationship between Gibbs measures on a subsystem and Gibbs measures on the original system. This can be described in two ways: combinatorially and analytically and the goal is to relate the two. Oseledets' famous multiplicative ergodic theorem gives a structural description of the action of a composition of a stationary sequence of random matrices on a Euclidean space. The space is decomposed into subspaces, each of which has a different asymptotic expansion rate. This work has been further developed and applied to certain families of quasi-compact operators in place of matrices. In ongoing collaborative work I propose to further develop this theory with a view to applications, in particular the application of the theory to environmental data sets.

所提出的研究涉及到动力系统的一些相互关联的方面（进化的研究）。 过程）和遍历理论（随机演化过程）。它还涉及概率论。的 主要的主题是（1）分段等距的行为;（2）热力学形式主义;（3） 算子值乘法遍历定理。 分段等距是不连续动力系统中最简单的例子。地图是 通过对欧几里得空间的子集进行划分并对每个部分应用等距来定义。由于 每个分区元素上的映射的简单性，无论这些系统中发生什么复杂性， 地图的不连续性。我们的目标是为此开发一个基础广泛的工具包和理论 映射类 在热力学形式主义中，吉布斯测度被定义为使量最大化的不变测度 类似于在热力学中产生的那些。这种情况的一个限制是遍历优化，其中一个 研究最大化潜力的措施。这在化学上对应于“将温度降低到 0'.该项目将研究这些最大化措施的一般性质。该领域的第二个主题是 理解子系统上的Gibbs测度与原始子系统上的Gibbs测度之间的关系 系统这可以用两种方式来描述：组合和分析，目标是将两者联系起来。 Oseledets著名的乘性遍历定理给出了一个 一个欧氏空间上随机矩阵的平稳序列的合成。空间被分解为 子空间，其中每个子空间具有不同的渐近展开速率。这项工作得到了进一步发展， 适用于某些家庭的准紧算子的地方矩阵。在正在进行的协作工作中， 建议进一步发展这一理论，以期应用，特别是应用的理论， 环境数据集。