Mathematical Sciences: Three Measures on Fractals

数学科学：分形的三种测度

• 批准号: 9302728
• 负责人: Alexander Volberg
• 金额: $ 5万
• 依托单位: Michigan State University
• 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 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302728&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Three; Measures; Fractals

This award supports mathematical research focusing on holomorphic dynamical systems in the plane, exploiting the interaction between classical function theory and ergodic theory. Particular emphasis is placed on thermodynamical formalism. Such systems are given by holomorphic functions defined in neighborhoods of compact sets defined to be the infinite past history of the neighborhood. Of the three objects of this work, the first is to establish the singularity of harmonic measure on the compact set with respect to one-dimensional Hausdorff measure. The second objective is finding the description of those cases where the harmonic measure is equivalent to the measure of maximal entropy. The third goal is to investigate the rigidity of harmonic measure in the sense that if two holomophic dynamical systems are quasiconformally conjugate with equivalent measures, can one determine if they are actually conformally conjugate? All of these problems have their roots in potential theory, combining probabilisitic and ergodic theoretic techniques. The research is related to analysis of fractal sets, since the most interesting compact sets formed by the infinite past of holomorphic functions are often of this type.

该奖项支持专注于平面全纯动力系统的数学研究，利用经典函数理论和遍历理论之间的相互作用。 特别强调热力学形式主义。 这样的系统由在紧集邻域中定义的全纯函数给出，紧集定义为邻域的无限过去历史。 在这项工作的三个目标中，第一个是在一维豪斯多夫测度的紧集上建立调和测度的奇异性。 第二个目标是找到调和度量相当于最大熵度量的情况的描述。 第三个目标是研究调和测度的刚性，即如果两个全息动力系统与等效测度拟共轭共轭，我们能否确定它们是否实际上共轭共轭？ 所有这些问题都源于势论，结合了概率和遍历理论技术。 该研究与分形集的分析相关，因为由全纯函数的无限过去形成的最有趣的紧集通常属于这种类型。