Dynamics and Ergodic Theory

拟议的研究涉及一些相互关联的方面的动力系统（evolvingprocesses的研究）和遍历理论（evolvingprocesses与随机性）。它还涉及概率论。主要内容有：（1）分段等距的行为;（2）热力学形式;（3）算子值乘性遍历定理。分段等距是不连续动力系统中最简单的例子。一个地图是通过采取一个分区的一个子集的欧几里德空间和应用等距的每一部分。由于每个划分元素上的映射的简单性，这些系统中发生的任何复杂性都是由于映射的不连续性。我们的目标是为这类映射开发一个基础广泛的工具包和理论。在热力学形式主义中，吉布斯测度被定义为最大化一个类似于热力学中出现的量的不变测度。一个极限情况是遍历优化，其中一个研究措施最大化的潜力。这在数学上相当于“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。在这方面的第二个主题是理解吉布斯措施之间的关系的一个子系统和吉布斯措施的originalsystem。这可以描述在两种方式：combinatorially和分析和目标是将两者联系起来。Oseledets著名的乘法遍历定理给出了结构描述的行动composition的一个固定序列的随机矩阵的欧几里德空间。空间被分解成子空间，每个子空间具有不同的渐近展开速率。这一工作得到了进一步的发展，并代替矩阵应用于某些拟紧算子族。在正在进行的合作工作中，我建议进一步发展这一理论，以期应用，特别是应用的理论toenvironmental数据集。

The proposed research concerns a number of inter-related aspects of dynamical systems (the study of evolvingprocesses) and ergodic theory (evolving processes with randomness). It also touches on probability theory. Theprincipal 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 isdefined by taking a partition of a subset of a Euclidean space and applying an isometry to each part. Due to thesimplicity of the map on each partition element, whatever complexity occurs in these systems happens becauseof the discontinuous nature of the map. The goal here is to develop a broadly-based toolkit and theory for thisclass of mappings.In thermodynamic formalism, Gibbs measures are defined to be invariant measures maximizing a quantityanalogous to those arising in thermodynamics. A limiting case of this is ergodic optimization in which onestudies measures maximizing a potential. This corresponds thermodynamically to 'lowering the temperature to0'. The project will study general properties of these maximizing measures. A second topic in this area isunderstanding the relationship between Gibbs measures on a subsystem and Gibbs measures on the originalsystem. 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 acomposition of a stationary sequence of random matrices on a Euclidean space. The space is decomposed intosubspaces, each of which has a different asymptotic expansion rate. This work has been further developed andapplied to certain families of quasi-compact operators in place of matrices. In ongoing collaborative work Ipropose to further develop this theory with a view to applications, in particular the application of the theory toenvironmental data sets.