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

Abstract: Let $K$ be a compact metric space and let $\gamma = (\gamma_1, \dots, \gamma_n)$ be a system of proper contractions on $K$. We study a C*-algebra $\mathcal{MC}{\gamma_1, \dots, \gamma_n}$ generated by all multiplication operators by continuous functions on $K$ and composition operators $C{\gamma_i}$ induced by $\gamma_i$ for $i=1, \dots, n$ on a certain $L^2$ space. Suppose that $K$ 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 $\mathcal{MC}_{\gamma_1, \dots, \gamma_n}$ is isomorphic to the Cuntz algebra $\mathcal{O}_n$ under some conditions.