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

Let $K$ be a compact metric space and let $\varphi: K \to K$ be continuous. We study a C*-algebra $\mathcal{MC}_\varphi$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_\varphi$ induced by $\varphi$ on a certain $L^2$ space. Let $\gamma = (\gamma_1, \dots, \gamma_n)$ be a system of proper contractions on $K$. Suppose that $\gamma_1, \dots, \gamma_n$ are inverse branches of $\varphi$ and $K$ is self-similar. We consider the Hutchinson measure $\mu^H$ of $\gamma$ and the $L^2$ space $L^2(K, \mu^H)$. Then we show that the C*-algebra $\mathcal{MC}_\varphi$ is isomorphic to the C*-algebra $\mathcal{O}_\gamma (K)$ associated with $\gamma$ under the open set condition and the measure separation condition. This is a generalization of our previous work, in which we studied the case where $\gamma$ satisfied the finite branch condition.