Dynamics and Ergodic Theory

拟议的研究涉及动力系统的一些相互关联的方面(对进化的研究 过程)和遍历理论(具有随机性的演化过程)。它还涉及到概率论。这个 主要主题是(1)分段等距线的行为；(2)热力学形式论；(3) 算子值乘法遍历定理。 分段等距是不连续动力系统的最简单的例子之一。地图就是 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的。由于 每个分区元素上映射的简单性，无论这些系统中发生什么复杂情况，都是因为 地图的间断性。这里的目标是为此开发一个基础广泛的工具包和理论 映射的类。 在热力学形式中，吉布斯度量被定义为使一个量最大化的不变度量 类似于热力学中出现的那些。这种情况的一个极限情况是遍历优化，其中一个 研究最大化潜力的措施。这在热力学上相当于将温度降低到 0‘。该项目将研究这些最大化措施的一般性质。这一领域的第二个主题是 理解子系统上的Gibbs度量和原始上的Gibbs度量之间的关系 系统。这可以用两种方式来描述：组合式和解析式，目标是将两者联系起来。 Oseledets著名的乘法遍历定理给出了一种结构化的描述 欧氏空间上平稳的随机矩阵序列的合成。空间被分解成 子空间，每个子空间都有不同的渐近展开率。这项工作得到了进一步的发展和 用来代替矩阵的某些拟紧算子族的。在持续的协作工作中， 建议进一步发展这一理论，以期应用，特别是将这一理论应用于 环境数据集。

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.