Mathematical Sciences: Measures, Dynamics and Dimensions

研究人员将继续他们的研究，涉及各种类型的度量--豪斯多夫、包装、吉布斯状态等；关联函数；以及关联维度。这些概念将在各种动力系统的背景下进行研究，例如应用于Julia集或将Jordan曲线描述为排斥曲线。他们将在某些几何对象和结构上使用这些技术，例如自仿射集合和自相似对象。这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在“动力学”的标题下，可以把一组抽象的变换如何作用于光滑空间的现代理论放在下面。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类的过程中与几何相联系。

The investigators will continue their research involving measures of various kinds-Hausdorff, packing, Gibbs state, etc.; associated functions; and associated dimensions. These notions will be studied within the context of various dynamical systems as applied to Julia sets for example or to characterization of Jordan curves as repellers. They will use these techniques with regard to certain geometric objects and constructions, e.g. self-affine sets and self-- similar objects. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way, ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.