Mathematical Sciences: Three Measures on Fractals

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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.