Mathematical Sciences: Measures, Dynamics and Dimensions

数学科学：测量、动力学和维度

• 批准号: 9303888
• 负责人: R. Daniel Mauldin
• 金额: $ 8万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303888&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Measures; Dynamics; Dimensions

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.

调查人员将继续他们的研究涉及各种措施-豪斯多夫，包装，吉布斯状态等；相关函数;以及相关的维度。这些概念将在各种动力系统的背景下进行研究，例如应用于朱莉娅集或作为驱避器的约当曲线的表征。他们将使用这些技术来处理某些几何对象和结构，例如自仿射集和自相似对象。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流进行分类时与几何接触。