C*-algebras associated with complex dynamical systems

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Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra Oᵣ associated with R as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra A = C(Jᵣ) of continuous functions on the Julia set Jᵣ of R. The algebra Oᵣ is a certain analog of the crossed product by a boundary action. We show that if the degree of R is at least two, then C*-algebra Oᵣ is simple and purely infinite. For example if R(z) = z² - 2, then the Julia set Jᵣ = [-2,2] and the restriction R: Jᵣ → Jᵣ is topologically conjugate to the tent map on [0,1]. The algebra Oz₂ ₋ ₂ is isomorphic to the Cuntz algebra O∞. We also show that the Lyubich measure associated with R gives a unique KMS state on the C*-algebra Oᵣ for the gauge action at inverse temperature (°R) if the Julia set contains no critical points.