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

Abstract

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.