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 OA is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set JR and the self-similar map γ satisfies the open set condition respectively, then the associated C⁎-algebras OR(JR) and Oγ(K) are simple and purely infinite. Let ΣA be the associated infinite path space for the 0-1 matrix A, then C(ΣA) is known to be a maximal abelian subalgebra of OA. In this paper we shall show that C(JR) is a maximal abelian subalgebra of OR(JR) and C(K) is a maximal abelian subalgebra of Oγ(K).