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 O R 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 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 O R 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 z 2 -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.