Mathematical Sciences: Three Measures on Fractals

该奖项支持专注于平面上全纯动力系统的数学研究，利用经典函数论和遍历理论之间的相互作用。特别强调热力学形式主义。这样的系统由定义在定义为邻域的无限过去历史的紧集的邻域中的全纯函数给出。在这项工作的三个目的中，第一个目的是建立紧集上调和测度相对于一维Hausdorff测度的奇性。第二个目标是找到调和度量等价于最大熵度量的情形的描述。第三个目标是研究调和测度的刚性，即如果两个全纯动力系统与等价的测度拟共形共轭，能确定它们是否真的共形共轭吗？所有这些问题都有其根源于势理论，结合了概率论和遍历论技术。这项研究与分形集的分析有关，因为由全纯函数的无限过去形成的最有趣的紧集通常是这种类型的。

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.