C^*-Algebras associated with complex dynamical systems

Abstract

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(J R ) of continuous functions on the Julia set J R 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) = z 2 - 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 OR for the gauge action at inverse temperature log(deg R), if the Julia set contains no critical points.