Maximal abelian subalgebras of C^⁎-algebras associated with complex dynamical systems and self-similar maps

We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A, rational functions R and self-similar maps γ on a compact metric space K, respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra O_A is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set J_R and the self-similar map γ satisfies the open set condition respectively, then the associated C^⁎-algebras O_R(J_R) and O_γ(K) are simple and purely infinite. Let Σ_A be the associated infinite path space for the 0–1 matrix A, then C(Σ_A) is known to be a maximal abelian subalgebra of O_A. In this paper we shall show that C(J_R) is a maximal abelian subalgebra of O_R(J_R) and C(K) is a maximal abelian subalgebra of O_γ(K).