Measures, Dimension and Dynamics

测量、维度和动态

• 批准号: 0400481
• 负责人: R. Daniel Mauldin
• 金额: --
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400481&HistoricalAwards=false
• 关键词: Measures; Dimension; Dynamics

DMS 0400481D Mauldin - M UrbanskiMeasures, Dimension and DynamicsUniversity of N TexasAbstractThe principal investigators propose to further advance the theory ofconformal systems as well as its applications and connections to areas of mathematics such as the number theory, geometric measure theory,probability theory, and complex analysis. Of special interest is thedevelopment of analytical tools required within the context of conformalgraph-directed Markov systems, infinite iterated function systems,iterations of transcendental entire and meromorphic functions andholomorphic endomorphisms. The proposed work involves graph-directedMarkov systems with infinitely many vertices and edges, interpretationsbetween counting functions, approximations and asymptotic limits, theproperties of various dimensions and measures within the context of graphdirected Markov systems. Applications involve Gauss' circle problem andSteinhaus' lattice problem, Diophantine approximation and self-conformalmeasures, return times, Kolmogorov's superposition theorem, the geometryand dynamics of the Fatou function, Gibbs measures for elliptic functions,critically pseudo non-recurrent elliptic functions, multifractal analysisof hyperbolic exponential functions, and multifractal analysis of Axiom Aholomorphic endomorphisms.

DMS 0400481D Mauldin-M Urbanski测量、维度和动力学N德克萨斯大学摘要主要研究人员建议进一步推进共形系统理论及其在数论、几何测度论、概率论和复分析等数学领域的应用和联系。特别令人感兴趣的是在合形图有向马尔可夫系统、无限迭代函数系统、超越整函数和亚纯函数的迭代以及全纯自同态的背景下所需的分析工具的发展。所提出的工作涉及具有无限多个顶点和边的图导马氏系统，计数函数之间的解释，近似和渐近极限，在图导马氏系统的背景下的各种维度和度量的性质。应用包括Gauss圆问题和Steinhaus格问题，丢番图逼近和自协调测度，返回时间，Kolmogorov叠加定理，Fatou函数的几何和动力学，椭圆函数的Gibbs测度，临界伪非递归椭圆函数，双曲指数函数的多重分形分析，以及公理全纯自同态的多重分形分析。