Asymptotic quantization errors for in-homogeneous self-similar measures supported on self-similar sets

We study the quantization for a class of in-homogeneous self-similar measures $\mu$ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for $\mu$ of order $r\in(0,\infty)$ and determine its exact value $\xi_r$. Furthermore, we show that, the $\xi_r$-dimensional lower quantization coefficient for $\mu$ is always positive and the upper one can be infinite. We also give a sufficient condition to ensure the finiteness of the upper quantization coefficient.