Measures, Dimension and Dynamics

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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.