Cartan subalgebras of C*-algebras associated with complex dynamical systems

Let $R$ be a rational function with degree $\ge 2$ and $X$ be its Julia set, its Fatou set, or the Riemann sphere. Suppose that $X$ is not empty. We can regard $R$ as a continuous map from $X$ onto itself. Kajiwara and Watatani showed that in the case that $X$ is the Julia set, $C_0(X)$ is a maximal abelian subalgebra of ${\mathcal{O}}_R(X)$, where ${\mathcal{O}}_R(X)$ denotes the C*-algebra associated with the dynamical system $(X,R)$ introduced by them. In this paper, we develop their result and give the equivalent condition for $C_0(X)$ to be a Cartan subalgebra of ${\mathcal{O}}_R(X)$.