C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches

Let K be a compact metric space and let phi : K -> K be continuous. We study a C*-algebra MC phi generated by all multiplication operators by continuous functions on Kappa and a composition operator C-phi induced by phi on a certain L-2 space. Let gamma = (gamma 1,...,gamma n) be a system of proper contractions on K. Suppose that gamma 1,...,gamma n are inverse branches of phi and Kappa is self-similar. We consider the Hutchinson measure mu(H) of gamma and the L-2 space L-2(Kappa, mu(H)). Then we show that the C*-algebra MC phi is isomorphic to the C*-algebra O-gamma(Kappa) associated with gamma under some conditions.