C* -algebras associated with complex dynamical systems

Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra script O sign_R associated with R as a Cuntz-Pimsner algebra of a Hubert bimodule over the algebra A = C(J_R) of continuous functions on the Julia set J _R of R. The algebra O_R 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 J_R = [-2,2] and the restriction R: J_R → J_R is topologically conjugate to the tent map on [0,1]. The algebra O_Z2-2 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 O_R for the gauge action at inverse temperature log(deg-R), if the Julia set contains no critical points. Indiana University Mathematics Journal