  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.
