Mathematical Sciences: Investigations in Conformal DynamicalSystems

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The two principal investigators will use the theory of one- sided Bernoulli shifts to understand the dynamics of maps on Cantor Julia sets for polynomials of arbitrary degree. Using the conjugacy between these two maps, they will relate the topology of the space of dynamical systems to the structure of the automorphism group of the one-sided shift. In addition, Teichmuller theory for punctured tori, classifications of dynamical systems which arise from iteration of rational maps on the Riemann sphere, and Catalan numbers for branched coverings by Riemann spheres will be the subjects of related investigations. Computer-generated fractal images can be made by analyzing the dynamics of polynomial maps of the complex plane. Pixels which continue to bounce about the plane, but never fall into one of the periodic attractors, comprise invariant fractals with exotic configurations. The mathematics behind these computer images requires an amalgamation of several existing theories. The principal investigators will continue their studies of these interrelationships.