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 $C^{\ast }$-algebra ${\mathcal{O}}_R$ associated with $R$ as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra $A = C(J_R)$ofcontinuousfunctionsontheJuliaset$J_R$of$R$\mathrm{.}Thealgebra${\mathcal O}R$ 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 $C^{\ast }$-algebra ${\mathcal{O}}_R$ is simple and purely infinite. For example if $R(z) = z^2 - 2$,thentheJuliaset$J_R = [-2,2]$andtherestriction$R : J_R \to J_R$ is topologically conjugate to the tent map on $[0,1]$. The algebra ${\mathcal O}{z^2 - 2}$isisomorphictotheCuntzalgebra${\mathcal O}_{\infty}$. We also show that the Lyubich measure associated with $R$ gives a unique KMS state on the $C^{\ast }$-algebra ${\mathcal{O}}_R$ for the gauge action at inverse temperature $\log (\mathrm{deg}R)$ if the Julia set contains no critical points.