Measures, Dimension and Dynamics

University Of North Texas, Denton TX

Investigators

Abstract

DMS 0400481 D Mauldin - M Urbanski Measures, Dimension and Dynamics University of N Texas Abstract The principal investigators propose to further advance the theory of conformal 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 the development of analytical tools required within the context of conformal graph-directed Markov systems, infinite iterated function systems, iterations of transcendental entire and meromorphic functions and holomorphic endomorphisms. The proposed work involves graph-directed Markov systems with infinitely many vertices and edges, interpretations between counting functions, approximations and asymptotic limits, the properties of various dimensions and measures within the context of graph directed Markov systems. Applications involve Gauss' circle problem and Steinhaus' lattice problem, Diophantine approximation and self-conformal measures, return times, Kolmogorov's superposition theorem, the geometry and dynamics of the Fatou function, Gibbs measures for elliptic functions, critically pseudo non-recurrent elliptic functions, multifractal analysis of hyperbolic exponential functions, and multifractal analysis of Axiom A holomorphic endomorphisms.