The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

Let $(g_i)_{i=1}^M$ be a family of contractive similitudes satisfying the open set condition. Let $\nu$ be a self-similar measure associated with $(g_i)_{i=1}^M$. We study the quantization problem for the in-homogeneous self-similar measure $\mu$ associated with a condensation system $((f_i)_{i=1}^N,(p_i)_{i=0}^N,\nu)$. Assuming a version of in-homogeneous open set condition for this system, we prove the existence of the quantization dimension for $\mu$ of order $r\in(0,\infty)$ and determine its exact value $\xi_r$. We give sufficient conditions for the $\xi_r$-dimensional upper and lower quantization coefficient to be positive or finite.