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 $μ$ 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 $μ$ of order $r\in(0,\infty)$ and determine its exact value $ξ_r$. Furthermore, we show that, the $ξ_r$-dimensional lower quantization coefficient for $μ$ 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.

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