Study on Complex dynamical systems and operator algebras

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In the research we constructed C*-algebras of operators on Hilbert spaces for complex dynamical systems on a complex plane by the iteration of meromorphic functions like rational functions and transcendental entire functions. We found unexpected relations between complex dynamical systems and operator algebras. Self-similarity of Julia sets of complex dynamical systems corresponds to purely infinite simplicity of the associated C*-algebras. The branch structure of complex dynamical systems is related with the structure of KMS states (equilibrium states) for the gauge actions on the associated C*-algebras. We showed that we can recover the degree of the rational function, the number of branched points and the number of exceptional points.