A bridge between complex analysis and operators on Hilbert spaces

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The aim of the research is to study a relation between complex dynamical systems and operators on Hilbert spaces. It is of fundamental interest in operator algebras to analyze interplay between a geometric or dynamical object and a C^*-algebra associated with it. For a branched covering, Deaconu and Muhly introduced a C^*-algebra associated with it using a r-discrete groupoid. A typical example of a branched covering is a rational function regarded as a self-map of the Riemann sphere. In order to capture information of the branched points for the complex dynamical system arising from a rational function R, we introduced a slightly different construction of a C^*-algebra O_R associated with R on the Julia set J_R. The C^*-algebra O_R is the Cuntz-Pimsner algebra of a Hilbert bimodule over the C^*-algebra C(J_R) of the set of continuous functions on J_R.The main result of our research is the following : For any rational function of R, If the degree of R is at least two, then C^*-algebra O_R is simple and purely infinite. Similarly we study a C^*-algebra O_γ associated with a system γ of proper contractions on a self-similar set. We also obtain the following result : If the system γ satisfies the open set condition, then C^*-algebra O_γ is simple and purely infinite.