Measures, Dimension and Dynamics

DMS 0400481 D Mauldin - M Urbanski测度、维数和动力学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应用和与数学领域的联系，如数论、几何测度论、概率论和复分析。 特别感兴趣的是thedevelopment的分析工具，需要在conformalgraph-directed马尔可夫系统，无限迭代函数系统，迭代的超越整个和亚纯函数和全纯自同态的背景下。所提出的工作涉及图导向马尔可夫系统与无穷多个顶点和边，计数函数之间的解释，近似和渐近极限，各种尺寸和措施的性质在图导向马尔可夫系统的上下文中。应用涉及高斯圆问题和Steinhaus格问题，丢番图逼近和自共形测度，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，临界伪非常椭圆函数，双曲指数函数的多重分形分析，以及Axiom Aholomorphic自同态的多重分形分析。

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.